Oofta
Legend
Shield Master is a Feat vs Battlemaster being a progressive Class Feature. But I see your point. You can argue that Maneuvers have a supernatural feel in that they can allow an element of fantastical capability to the fighter. But that's more flavor than mechanic. It's like when Captain America bounces his shield to hit three Hydra Agents. You could say that's not really possible (supernatural) but he's just that good.
But you never hear Cap say "Sorry Iron Man, I can't throw my shield right now because I don't have any superiority dice left".
That's different from someone running out of ammunition (or spell slots), or the enemy agents aren't in correct formation for the shield to bounce correctly. It's a completely artificial, and not in any way natural, limitation.
I could see some type of mechanism involving usage of stamina, only being allowed under certain circumstances, or similar mechanisms. My rule of thumb is: would this make sense in an action movie or a book? Most mechanisms in D&D meet that criteria, maneuvers do not.
Barbarians are supernatural warriors, powered by animal totems, spirits of their ancestors, etc. Monks have a reserve of supernatural energy in the form of ki. Saying that battlemasters have an artificial, unnatural mechanism isn't necessarily a bad thing.