Antibiotics are actually an exception to my general opinion that all medications should be available without prescription. Unlike most drugs, their use has major externalities which means there is a role for larger societal regulation of their use.

Also, are farms actually the major vector for antibiotic resistance in the human population? I was under the impression that the majority of antibiotic resistant infections occur in places like hospitals rather than among farm workers, which would seem to indicate farm animals are not the main problem (I 100% support banning the practice anyway).

I'd ask the opposite: are hospitals the places where the immunities develop or are they merely the places where the presence of antibiotic resistant bacteria is most deadly and therefore investigated?

The dangerous thing about antibiotic immunity is that it can transfer also between bacteria species.