It does not help much

One girl and the DM in one of the groups I played in were also big fans of Avatar o último dobrador de ar, so they tried everything and its opposite to make Wot4E appealing, with much difficulty.

They implemented something very similar to your proposal:
Além de the usual prorgression, she gained a new discipline every odd level. We played on level 12 when she tried this, so this gave her 7 instead of your 6 disciplines.
They were still seriously underwhelmed.

They even tried giving her todos the disciplines, still she was the weakest in the group¹

The problems of the Wot4E Monk

As she put it, "if I wanted to cast spells ao invés of Monking, I would have brought a Sorcerer".

With the sole exception of Fangs of the Fire Snake, the disciplines do not complement, but substituir your signature monk combat abilities (Martial Arts and Flurry of Blows).

Even that they do badly:

• Fireball is quite strong on level 5, but rather lackluster on level 11
• Fist of Unbroken Air costs 6 Ki points just to do as much damage as a Flurry of Blows for 1 Ki point². An Open Hand Monk can knock prone an enemy with that 12 times per short rest, but you can only do it twice. From 30 feet, but still a very steep price
  • This math encourages you to pour all your Ki points into one big attack, which is just the opposite of sane resource management

How we fixed it

Magias

With every spell³ you can spend 1 Ki point to use Martial Arts, 2 to use Flurry of Blows before or after the casting. While this is way behind what a Sorcadin can do, you will feel at least monkish when you use Ride the Wind to hit a Djinn twice.

This might be too strong on low levels, when your casting abilities are not também far behind a full caster.
On level 4 you can use Fist of Four Thunders to prone (and mildly damage) some enemies, followed up by Flurry of Blows, with advantage. This is probably better than anyone could do on this level, but now you just spent todos your resources, and cannot do anything special until a short rest. Next level you are again outshined by most classes⁴. But if you will, you can limit this ability to 5th level and above.

The player of the Monk did propose to make this all 1 Ki point cheaper, so 1 for Flurry of Blows, 0 for Martial Arts. The DM shut it down, arguing that in this case it would be as good as what a Valor Bard gets on level 14 or an Eldritch Knight on 18. I would have allowed it, for two reasons:

1. Martial Arts is weaker than a normal attack, it caps at 1d10+5, while an Eldritch Knight can easily get 2d6+2d6+5 (Flame Tongue Greatsword)
2. Mesmo se this were as strong as a normal attack, it does not make it overpowered, just adequate

In this case however, I would definitely limit it to level 5 and above.

Control disciplines

Basically Fist of Unbroken Air and Water Whip.
They became bônus actions, using d4s instead of d10s for damage. So pouring Ki into them is a bad idea, they are just for control.

2 Ki points for 3d4 damage might seem weak, and it is pitiful. However, you pay these Ki points for the best positional control in the game: pulling 25 feet is ótimo⁵. The damage is just gravy.

How many you know

After all these changes this is really a minor issue, but I think your suggestion is fine, double the number.

Cantrips

Cantrips are weak, so it does not matter much if you get any.

1) It was an optimized group, a Sorcadin, a Divine Soul Sorcerer/Warlock, a Fighter/Rogue and her
2) Flurry of Blows: 4*(1d8+5) = 38 damage, 7*(1d10) = 38.5
3) "Some elemental disciplines allow you to cast spells" (PHB p80)
4) Extra Attack and 3rd level spells are aquele forte
5) Pushing 25 feet is rarely so useful, unless you have a portable lava pit.

The Problem is Efficiency, not Versatility

The biggest problem with the Path of the Four Elements monk is efficiency, not versatility. This can be broken down as follows:

1. Four Elements Disciplines all compete with Stunning Strike, Flurry of Blows, Patient Defense, Step of the Wind, Deflect Missiles, and Diamond Soul for Ki use.
2. In terms of damage, the Four Elements monk will be the more effective when using their pre-existing Monk abilities instead of spells. Even using Fangs of the Fire Snake, you are burning through Ki at a breakneck pace (finalmente two Ki) to pick up a couple d10 damage.
3. When the Four Elements Monk gains spells via disciplines, these spells are not Ki-efficient. Whereas the Shadows monk can cast spells at a 1 Ki = 1 Spell Level conversion rate, the Four Elements monk ends up spending more for the same effect (usually 1 Ki = Spell Level + 1).
4. Many of the spells on the Four Elements discipline list compete directly with other monk abilities (especially Patient Defense, Flurry of Blows, and Stunning Strike) in the action economy as well. If your monk casts Pessoa em espera, they are missing out on finalmente three attacks, or four attacks with Flurry of Blows.
5. All of the Four Elements Monk disciplines use Ki, and the Four Elements monk gains no relevant abilities when they are not using Ki. Despite this, the Four Elements monk gains no bonuses to their Ki progression. Due to the crappy monk Ki progression compared to the expensive Four Elements disciplines, they are likely to burn through all of their Ki before they do anything useful. No other monk subclass suffers this problem. All of them have abilities which make their existing abilities more efficient or have abilities which work without expending Ki.
6. Most of the spells that the Four Elements monk gets are garbage by the time you get them. Fireball, for example, is utter rubbish at Level 11. And at Level 17, the Wizard gets Desejo, whereas you can cast Muro de pedra for 6 Ki. This is thoroughly unimpressive.

What the class needs is to expand their Disciplines list to include some selections which are unique and interesting and which do not use Ki. Furthermore, the Four Elements monk should be able to cast spells at a 1 Spell Level = 1 Ki conversion rate, so that the Disciplines are actually worth using. Increasing the number of disciplines learned is a tertiary issue.

Cantrips are mostly irrelevant to a primary warrior class. The monk has excellent mobility, a good damage die, and gets to add their primary attack stat to their regular attacks. They will rarely, if ever, use an attack cantrip even if you give them one. Also, even if they got todos of the disciplines, this wouldn't matter much if they can't actually use them effectively. They need their Ki use to be more efficient, and to gain class abilities which do not require them to expend Ki.

I am currently playing a Four Elements monk using this homebrew remaster by SpiketailDrake on Reddit. My damage is quite good and it plays out in an interesting way (we're currently level 7), but I'm hardly overpowered. Even with this remaster, taking multiple "free" abilities, I burn through all of my Ki in no time.

I have attempted to analyze this from a few different angles to answer what I think the underlying question is:
Does a Way of the Four Elements Monk (Wot4E Monk) become unbalanced by virtue of having more disciplines accessible to them?

Unbalanced compared to what?

Comparison basis: Full, one/half, and one/third casters plus Warlocks

I started by continuing what you mentioned in your question, which is to attempt to discern what the Wot4E Monk gets from their feature compared to others (specifically Ranger, Eldritch Knight, and Sorcerer). To do this, I used the DMG's standard assumptions regarding adventuring days, which generally includes 2 short rests between long rests (pg. 84) with 2 encounters for each short rest (6 encounters per day). I also extended the comparison to the Warlock since there are a lot of similarities between Ki and Pact Magic, both of which are recovered on a short rest. Finally, I considered what the trade-off is when taking advantage of the extra options.

I've played Monks before, but never a Wot4E Monks due to the issues I perceive with it. This is an analytical review sans feelings and suspicions on the tradition.

A review of their disciplines shows the following which I used for guidance:

Tier 1 is roughly in balance (Ki usage has opportunity costs)

Low Ki point pool limits the impact of this change.

• Within 1st tier, they can learn Onda de trovão or Mãos ardentes both 1st level close range AoE damage spells; they can also learn Rajada de vento, a 2nd level battlefield control spell that doesn't include a mechanism for direct damage. Furthermore, within 1st tier (caps at level 4) they have the ability to cast these spells up to 2 times per short rest, so 6 times per day. Casting those spells exclusively means they will not have enough Ki to use Deflect Missiles' returning fire aspect or the other Ki features; the need to choose between the discipline spell or the other Ki features is a pretty big blow since that includes things like Patient Defense and Flurry of Blows.

  At 4th level, 1/2 and 1/3 casters have only 3 first level spells per day, so the Wot4E Monk is outpacing them right now; those casters know 3 spells and 4 spells, respectively (it is super weird that 1/3 casters know more spells than the 1/2 casters but that's a thing I learned today). Full casters have 4 first level spells and 2 second level spells, so they're about even; full casters know 5 spells. Warlocks have 2 second level spell slots and per my criteria could use them 6 times per day, so they're probably a bit ahead; Warlocks also know 5 spells.

  Within 1st tier, your proposal doesn't seem unreasonable. It would simply offer Wot4E Monks more options to expend their Ki at the expense of using their Ki for something else. Furthermore, the additional choices don't seem to provide exceptional utility that new options are available to the Wot4 Monk that weren't already present.

Tier 2: roughly in balance (Ki usage has opportunity costs)

The proposal begins to open up a play style difference from other monks.

• Within 2nd tier, they can learn Pessoa em espera, a 2nd level spell that's potentially deadly control or completely worthless depending on the enemy type; they can also learn Shatter a ranged AoE damage spell. Furthermore, within 2nd tier (caps at level 10) they have the ability to cast these spells up to 3 times per short rest with an extra Ki point for other uses, so 9 times per day (assumes no upcasting). Casting those spells exclusively means they will only have enough Ki for 1 other Monk feature like Deflect Missiles' returning fire aspect, Stunning Fist, or the other Ki features. The loss of the other Ki features is likely less felt in this tier since Wot4E Monk has some strong distance options, but the limited use of Stunning Fist for close combat will definitely be felt by the rest of the party. This pool is also going to be shared by the disciplines selected previously if appropriate.

  At 10th level, 1/2 casters have a 4, 3, 2 spread across 1st to 3rd level spells per day, so the Wot4E Monk is probably about even; those casters know 6 spells. 1/3 casters have a 4, 3 spread across 1st to 2nd level spells per day, so the Wot4E Monk is outpacing them; and curiously they know 7 spells despite 1/2 casters knowing one less. Full casters have a 4, 3, 3, 3, 2 spread across 1st to 5th level spells, so they're way ahead; full casters know 11 spells. Warlocks have 2 fifth level spell slots and per my criteria could use them 6 times per day, so they're probably a bit ahead; Warlocks also know 10 spells.

  Within 2nd tier, your proposal doesn't seem unreasonable. It does appear that Wot4E Monk might start to show more preference for distance that isn't normally seen in other types of Monks at this tier and your additional options will help them to maintain that distance. The additional selections don't seem to provide exceptional utility that weren't already available to the Wot4 Monk, you just give them more opportunities to find.

Tier 3: roughly in balance, playstyle changes (Ki usage has opportunity costs)

The "ranged more than melee" feel of this class becomes pronounced.

• Within 3rd tier, Wot4E Monk can learn Fireball a 3rd level ranged AoE damage spell; they can also learn Forma Gasosa e Voar, with the restriction to only target themselves. Both 3rd level buff spells are complicated by needing Concentration. Furthermore, within 3rd tier (caps at level 16) they have the ability to cast these spells up to 4 times per short rest, so 12 times per day (assumes no upcasting). Interesting combinations start occurring in this tier that weren't previously available, specifically in the form of a Wot4E Monk that takes Fireball and Fly; they can now play a very deadly ranged game, potentially using the two spells to fire off 3 long-distance fireballs into enemy groups with limited risk of reprisal.

  Casting those spells exclusively means they will not have enough Ki for other Monk feature like Deflect Missiles' returning fire aspect, Stunning Fist, or the other Ki features. The loss of the other Ki features is likely less felt in this tier since Wot4E Monk has a very heavy duty distance option, but the limited use of Stunning Fist for close combat will definitely be felt by the rest of the party. This pool is also going to be shared by the disciplines selected previously if appropriate.

  At 16th level, 1/2 casters have a 4, 3, 3, 2 spread across 1st to 4th level spells per day, so the Wot4E Monk is probably ahead again; those casters know 9 spells. 1/3 casters have a 4, 3, 3 spread across 1st to 3rd level spells per day, so the Wot4E Monk is outpacing them; and curiously they know 11 spells despite 1/2 casters knowing two less. Full casters have a 4, 3, 3, 3, 2, 1, 1, 1 spread across 1st to 8th level spells, so they're way ahead; full casters know 15 spells. Warlocks have 3 fifth level spell slots and per my criteria could use them 9 times per day and also have Arcanum through 8th level, so they're probably a bit ahead; Warlocks also know 13 spells plus their Arcanum.

  Within 3rd tier, your proposal seems to substantially alter the Wot4E Monk, but it's probably ok.. This tier gains a lot out of an additional discipline that wasn't previously available for Wot4E Monk in the form of having a very powerful nuke option. Much of this stems from the option to take Voar com Fireball, but this might not actually be a serious issue considering what an unaltered Wot4E Monk would have; assuming a human the Monk has a land speed of 55' and can pretty much ignore vertical surfaces provided they end their turn on flat ground, this is only 5' slower than a creature with a fly speed granted by Voar. Finalmente, Voar has the Concentration issue tied into it, so if the Monk eats a good hit, this entire thing might disappear before it really gets going. This tier will see a continuation of the trend for the Wot4E Monk to be more distance oriented than melee.

Tier 4: roughly in balance, playstyle changed (Ki usage has opportunity costs)

There isn't enough Ki for this option to overpower other options.

• Within 4th tier, Wot4E Monk can learn Cone of Cold a 5th level AoE damage spell; they can also learn Wall of Fire a 4th level AoE damage and control spell; they can also learn Wall of Stone, a 5th level control/utility spell. Furthermore, within 4th tier (caps at level 20) they have the ability to cast these spells between 3 and 4 times per short rest, so 9-12 times per day (assumes no upcasting). This tier actually looks a lot like 2nd tier for the Wot4E Monk, some damage option and a control option but nothing that really synergizes exceptionally well to make me question that there might be an issue.

  Casting those spells exclusively means they will not have enough Ki for other Monk feature like Deflect Missiles' returning fire aspect, Stunning Fist, or the other Ki features. It also means that Empty Body is not available, either, and that is a fairly significant loss considering that feature's potency for improving survival at this tier. This pool is also going to be shared by the disciplines selected previously if appropriate.

  At 20th level, 1/2 casters have a 4, 3, 3, 3, 2 spread across 1st to 5th level spells per day, so the Wot4E Monk is ahead again; those casters know 11 spells. 1/3 casters have a 4, 3, 3, 1 spread across 1st to 4th level spells per day, so the Wot4E Monk is outpacing them; and curiously they know 13 spells despite 1/2 casters knowing two less. Full casters have a 4, 3, 3, 3, 2, 2, 2, 1, 1 spread across 1st to 9th level spells, so they're way ahead; full casters know 15 spells. Warlocks have 4 fifth level spell slots and per my criteria could use them 12 times per day and also have Arcanum through 9th level, so they're ahead; warlocks also know 15 spells plus their Arcanum.

  Within 4th tier, your proposal doesn't seem unreasonable. The additional selections don't seem to provide exceptional utility that weren't already available to the Wot4E Monk, you just give them more opportunities to find.

Resumo

Ki pool limitations prevent loss of balance; Wot4E Monks like to fight more from a distance than other types of monks and as a result, don't get to utilize a lot of the other monk features. Your proposal doesn't really alter this, it actually enhances it by giving the Wot4E Monks more opportunities to keep their distance. A party with a Wot4E Monk needs to recognize that they won't be seeing a litany of Stunning Strikes because their Monk is going to try and be more or less of a 1-trick pony with some sort of distance spell.

I think that may be the crux of the issue for the base Wot4E Monk. They will play like a much more limited Sorcerer because most of their disciplines work best from a distance, which means they will gain class features that aren't practical for them to implement. I think your proposal will alleviate some of that one-trick pony issue, but being a two-trick pony isn't much better.

You also asked about adding a few cantrips that are thematically appropriate. Some are ranged attack based in nature and I think will further reinforce the ranged nature of the Wot4E Monk. The others are more flavorful or highly situational and I don't see an issue with them.