One bird that has cropped up quite a lot in discussions of aviary birds are the turacos but they are quite interesting in that, although they would need an aviary to feel realistic, I don't think they would need full flight in the way that, say, a parrot or bird of prey might. This is a direct quote from the turaco best practice guidelines by the EAZA:
"Turaco are not built for flying distances or sustained flight, their flight appears weak and quite arduous. They are much more at home running along tree branches and hopping through vegetation, they are equipped with semi-zygodactylous toes, giving them impeccable arboreal locomotion."
I wonder that, if they come to the game at all, they would almost be better off being animated like a more standard arboreal animal. They would hop and run about when on the ground but would spend a lot of time running around on climbing frames, with a glide or a clumsy flapping flight replacing the jumps.