Your question = Why do Americans who feel the UFC isn't fair just go to Bellator then? You know the answer.
Would you guys be happy with less competition? Weird to me.
A few have have left the UFC because they felt it the UFC was unfair (most famously, Randy Couture tried to do just that, though he ran into difficulties because he had fights left on his contract). And a few fighters around the world, notably Fedor, never signed up with UFC in the first place because they thought its contract was unfair. Not going to the UFC is a definite possibility, there are other options, both in America and around the world, including in Brazil.
In any case, my point isn't that Brazilians (and others) should leave the UFC, but that it isn't as unfair as suggested, given that not many are leaving. The reason for fighters going to the UFC, and remaining after a contract is finished, is that they feel its the best option.
And actually I'm not arguing that the UFC is fair - you need only look at Couture and Fedor (an American and a Russian) to see that many who have had dealings with it think its not. GSP seems to be on the list of those who think its not completely fair either, given he's pushing for a fighter's union. But for all that its not perfect, its clearly acceptable to most fighters, including the Brazilians in it, who keep renewing their contracts.
In fact its dominance lies in its ability to attract and keep most of the best fighters. If the fighters thought it was fundamentally unfair, they'd move on, and the UFC would drop in importance.