Perhaps because there are subsidies based on providing broadband/fiber service to specific areas.

The FCCs mandate is to protect the airwaves from interference. Starlink uses shared bands. if the receivers move, tracking down interference is harder.

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.

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

Surveillance & Monopoly Protection

If people stopped using cellphone service for mobile internet they would loose their immense location and surveillance capabilities (gov).

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.