Do they have authority to impose such limitations? It sounds very overreaching.

What makes more sense is probably Starlink trying to prevent uneven network load.

What about mobile networks which literally do just that? Or they don't use shared bands?

this is not really true relative to what OP said. there are fcc regulations for spectral leakage and such, but that's well known and means you can't transmit outside a service area. but that just means you're in another, adjacent service area. it has more to do with the spectral efficiency going down as you move from boresight, so they don't want a capacity loss. fcc has nothing to do with this decision.