I like the idea of the malleable approach to the toolbox of D&D advocated in the OP (i.e. an owlbear as a unique creature, not necessarily a member of a species). On the other hand, I don't think a DM "must" allow everything the players want, no matter how much it clashes with their setting premise and themes.
One thing that often gets lost in the mix is the enjoyment of the DM. In some sense, the DM's enjoyment trumps all: if they're not having fun, chances are no one will, and the DM not having fun is more challenging to troubleshoot than an individual player not getting their double katana-wielding kenku.
In that sense, I think the DM owes it to themself--and the group--to run a world that they want to run. Their job isn't simply "entertain the players" or be a font of instant wish-fulfillment; it is to orchestrate an experience that is entertaining for all, and sometimes that involves restraint and holding back on one wish to further a larger plan (not unlike telling your kids why they can't eat ice cream for every meal).
It isn't either/or, of course, and really there's a Goldilocks zone - but it is pretty broad, unless a player gets fixated on a specific thing and/or the DM's setting and game is very thematically narrow.
I think also that part of the social contract of D&D is that the players are, in a sense, "guests" in the DM's "house." The DM puts a lot more work into the game, and that should be honored. Now if the DM says, "The theme of the world is My Little Centaur" and no one is into it, that's just how it goes and the DM has to accept that, unless they want to find a group that is into it. But that sort of thing is the exception. If the DM is saying, "no warforged," I think it behooves the players to accept that, and not feel entitled for the DM to explain their reasons why beyond "It doesn't fit the overall themes I'm going for." There, done - no further explanation required, and a reasonable human being will move onto something else (maybe dual katana-wielding kenku).