I think the premise of the question is not necessarily correct. There are not actually that many magical creatures that live in the forest (at least that we see throughout the books).
The following magical creatures are seen in the forest:
- Centaur herd
- Thestral herd
- Acromantula colony
- Bowtruckles
- Unicórnios
- Hippogriffs
Centauros
De acordo com a Fantastic Beasts e onde encontrá-los:
Wizarding authorities in each of the countries where centaurs are found have allocated areas where the centaurs will not be troubled by Muggles;
A forest on the grounds of a wizard school hidden from muggles is the perfect place to be an "allocated area". Thus it is unsurprising to find centaurs in the Forbidden Forest and not in any random forest in Britain.
Additionally, the other forests in which centaurs are found may have Muggle-repelling charms, as mentioned in the introduction to Animais fantásticos, so you wouldn't be able to find them:
Muggle-repelling charms prevent trespassers into the forests where centaurs and unicorns live and on the lakes and rivers set aside for the use of merpeople.
And even without that, the introduction also states that centaurs deliberately avoid Muggles:
Then there are those beasts that, due to cleverness or innate shyness, avoid contact with Muggles at all costs — for instance, the unicorn, the Mooncalf and the centaur.
Thestrals
Hagrid explains in Ordem da Fenix why they have a thestral herd at Hogwarts:
"Yeah, I was gonna tell yeh how come we got a herd. Yeah, so, we started off with a male an' five females. This one," he patted the first horse to have appeared, "name o' Tenebrus, he's my special favorite, firs' one born here in the forest —"
This makes it sound like the thestrals were deliberately introduced into the forest. The reason you wouldn't find thestrals in any random forest in Britain is that they are rare, as described in Animais fantásticos:
and the rare Thestral (black, possessed of the power of invisibility and considered unlucky by many wizards)
Acromantula
Acromantula are not native to Britain. As Aragog tells Harry and Ron in Câmara de segredos:
"I was not born in the castle. I come from a distant land.
And as confirmed in Animais fantásticos:
It originated in Borneo, where it inhabits dense jungle.
The only reason they are found in the Forbidden Forest is that Hagrid introduced a male and female there, and they had lots of offspring. In fact, we can surmise that they are pretty rare because Animais fantásticos também diz:
The beast is believed to be wizard-bred, possibly intended to guard wizard dwellings or treasure, as is often the case with magically created monsters.
Thus, the only Acromantula in existence are probably the limited number that were intentionally created, and their offspring. It is certainly understandable why you would not find them in a random forest in Britain.
Additionally, the introduction to Animais fantásticos states that the areas where Acromantula are found are inaccessible to Muggles:
Other magical creatures inhabit places inaccessible to Muggles — one thinks of the Acromantula, deep in the uncharted jungle of Borneo,
Bowtruckles
Bowtruckles probably pode be found in random forests in Britain. As per Animais fantásticos:
The Bowtruckle is a tree-guardian creature found mainly in the west of England, southern Germany and certain Scandinavian forests.
However, you may not have ever noticed them in your travels through various forests because they are very hard to notice, again, as per Animais fantásticos:
It is immensely difficult to spot, being small (maximum eight inches in height) and apparently made of bark and twigs with two small brown eyes.
Unicórnios
There does not seem to be evidence that unicorns were deliberately introduced into the Forbidden Forest. According to Animais fantásticos:
The unicorn is a beautiful beast found throughout the forests of northern Europe.
As such, it might be possible to find them in a random forest in Britain. However, even in a forest in which unicorns can be found, you might not actually see them, because according to Animais fantásticos:
It generally avoids human contact, is more likely to allow a witch to approach it than a wizard, and is so fleet of foot that it is very difficult to capture.
Additionally, it is stated in the introduction to Animais fantásticos that unicorns have territories designated for them (of which the Forbidden Forest might be one):
While unicorns and merpeople are only too happy to stay within the territories designated for their use,
Not only that, there may even be muggle-repelling charms on the forests where unicorns live, as also mentioned in the introduction:
Muggle-repelling charms prevent trespassers into the forests where centaurs and unicorns live and on the lakes and rivers set aside for the use of merpeople.
And also mentioned there, unicorns avoid Muggles at all costs:
Then there are those beasts that, due to cleverness or innate shyness, avoid contact with Muggles at all costs — for instance, the unicorn, the Mooncalf and the centaur.
Hippogriffs
It's unclear whether hippogriffs actually live in the Forbidden Forest. We never actually see any hippogriffs in the forest. The only reason to think there are hippogriffs in the forest is that Hagrid leads them out of the forest for the first Care of Magical Creatures lesson in Prisioneiro de Azkaban:
"Righ' then," said Hagrid, who seemed to have lost his thread, "so — yeh've got yer books an' — an'— now yeh need the Magical Creatures. Yeah. So I'll go an' get 'em. Hang on..."
He strode away from the forest and out of sight.
However, it is possible that Hagrid simply stored them in the forest before the lesson so they would be a surprise. In fact it is possible that Hagrid only had the whole group of hippogriffs just for the lesson (e.g. he rented them for a few days or something). Indeed, throughout the rest of the series there is no mention of any of the other hippogriffs — not in the forest, nor outside of the forest. The only hippogriff we do continue to see is Buckbeak, and both in Prisioneiro de Azkaban, and when he returns in Príncipe Mestiço, Buckbeak lives in or outside of Hagrid's cabin.
Also, we don't even know if hippogriffs normally live in forests. Unlike the other creatures, there is no mention in the hippogriff entry in Animais fantásticos that they live in forests.
In short, there are not actually that many magical creatures that we ever see in the Forbidden Forest to begin with. And for each creature in the Forbidden Forest there is an explanation for why it would be found in the Forbidden forest and not in any random forest in Britain (with the possible exception of hippogriffs, which we don't even know with certainty resided in the Forbidden Forest).