Why didn't Harry Potter play as the king?

There is no good answer

Let's start with this, because it's true. There is no perfect answer. I just reread the first Harry Potter book a couple days ago, and there is no answer in the book that gives us insight into Ron's reasoning. The very best we can do is guess or make wild theories. It's important to keep context in mind, that we should be looking for reasonable answers, and basically all answers will be opinion based, or conjecture.

Fora do Universo

This seems like a plot hole. One of those things that, when I read the question, I said, "Yeah... That is kinda weird. It seems like the obvious choice."

1. It may be that J.K. Rowling simply overlooked this option.
2. Perhaps it seemed obvious to Rowling. There are a lot of "rules of magic" that we never actually see in the books, like all the "complex diagrams" of how magic actually works. It may be something simple like, "Well, they obviously não podia be the king. It's against the rules of McGonagall's game".
3. It makes Ron even cooler. Not only does he beat McGonagall's chess set, but he beats her set with three pieces he cannot lose. If Harry is king, it takes away from how good Ron is.
4. The editors/Rowling may not have wanted to make such a cliche moment as Harry taking the place of the King. Like, yeah, we get it, he's the main character. It would seem pretty asinine to make him the King piece.

In Universe

There have been a couple in-universe answers already, and I'll address them briefly.

1. They have to "Play their way across the board". This is possible, and does at least have one cited source in the material. However, it seems unlikely, as Ron seems to play the whole game to win, rather than to make a path. It seems like the kind of detail Rowling would have added had that been the plan. I think that "play our way across" was a turn of phrase, simply meaning that they have to win to go on.

2. The chessmen talk back. Ron is a wizard who grew up with wizard's chess. While McGonagall's chessmen don't seem to talk, Ron may be expecting them to give him a certain amount of pushback on any decision he makes. Therefore, he put Hermione and Harry in important positions to help the game. This seems like it wouldn't be the case either, however, and for a variety of reasons. Why wouldn't Ron make them more important pieces? The Queen, for example, if he needed troops that would obey him. If your Queen doesn't listen to you, you generally can't win a game of chess. Furthermore, yes, wizard chessmen backtalk, but they don't disobey. The problem Harry describes playing wizard's chess is that they try and convince him of making different moves than what he would normally. But Ron wouldn't have this problem. He is a veteran player, and he wouldn't take the word of his pieces even if they fez talk back.

Então, o que seria make sense? Well, there are a few in-universe answers that might fit an 11 year old chess master.

1. Harry and Hermione are a Rook (castle) and Bishop, on the same side of the board, and on the opposite side of Ron. It's possible he planned on keeping one side of the board as protected as possible to keep Harry and Hermione safe. Not only are they on the "safe side" in this idea, but they are two very mobile pieces. The Rook and Bishop can both go from one end of the board to the other in one move, so they can escape danger as needed.

2. The bishop and the rook were option 2. It's possible that Ron querido Harry to be king, but the King is irreplaceable. The transaction where Ron wanted this was cut from the book because it didn't really matter. Only the most critical SE questioners would ever really care. Logical fragments are often cut from kids books, as kids tend not to care as much.

3. Ron may have cared more about his friend's safety than for beating Voldemort. When the king is checkmated, they lose, but maybe the king gets bludgeoned to death upon losing. While it makes the game harder, Ron might have decided to play in a way where he kept Harry and Hermione safe above all other concerns. As previously explained, Harry and Hermione are very mobile pieces. He may keep them safe, and if it came to it, he would let the King be taken. (By the time you lose half your power pieces, you generally lose anyway).

   Remember that Fluffy and the Devil's Snare are both letal. While it's a fun kids book for us, Ron has just escaped back-to-back lethal situations, so he has no reason to believe the chess interactions won't also be deadly. (Makes his sacrifice much cooler, really, as he thought he was going to die to stop Voldemort. Maybe he deserved more than 50 points).

So, there isn't a good sourced interview or quote, but there options for you to choose to believe.

There's not much to go on, but here's Ron's initial assessment of the task:

"It's obvious, isn't it?" said Ron. "We've got to play our way across the room."

Behind the white pieces they could see another door.

"How?" said Hermione nervously.

"I think, " said Ron, "we're going to have to be chessmen."

(Emphasis mine.) A possible interpretation of this was that Ron's initial plan was to simply move from one side of the board to the other. Once they got past the white pieces, they could step off the board and get to the door. Since the king can only move one space at a time (and the queen is too important to not use against the white pieces), Ron picks middle-of-the-road pieces to replace, pieces he thinks he can get over to the door.

Later, due to the white team's successes, it seems that the plan changes, and Ron realizes the only way they're going to get through is to take out the king.

There was no alternative.

Keep in mind also that they're racing against the clock, and Ron is an eleven-year-old boy unused to the stresses of facing deadly chessmen in the pursuit of a dark wizard. So maybe he made a lapse of judgement, or didn't think his plan all the way through.

As for why Harry and Hermione didn't object:

"Now, don't be offended or anything, but neither of you are any good at chess--"

"We're not offended" said Harry quickly. "Just tell us what to do."

They trust that Ron knows what he's doing. And at the end of the day, he faz get them through the door.

We know that wizarding chess is different from Muggle chess from descriptions of Harry playing Ron over the winter holiday. In wizarding chess, the pieces are actually difficult to control, and skill at controlling them plays a part in the outcome of the game. We know that one of the reasons why Ron won against Harry so often was that Ron had inherited chess pieces that trusted him, but Harry had to borrow a chess set and the pieces often argued back.

In Chapter 12 of the Philosopher's Stone :

Ron’s set was very old and battered. Like everything else he owned, it had once belonged to someone else in his family – in this case, his grandfather. However, old chessmen weren’t a drawback at all. Ron knew them so well he never had trouble getting them to do what he wanted.

So, one interpretation is that Ron might have picked pieces that make the most moves on purpose, since his friends trust him and will follow his directions but the chessmen might not.