It occurs to me that not only is Bayes theorem more obviously correct than induction, it is also more general than induction.

Bayes theorem applies to all cases of updating beliefs upon receipt of evidence.

Induction is limited to a subset collection of cases - specifically those cases in which we wish to update our beliefs about a population using evidence which consists of a sample drawn from that population.

Edit to reply to your edit: Yes, I think that it is true that for many problems Bayes theorem isn't useful, and that for all problems where induction works, it is the fact that induction does work that makes Bayes theorem useful. These are all cases of updating based on a sample from a population. But there clearly are also problems where Bayesian reasoning is useful, but induction just doesn't apply. Problems where there is no population and no sample, but you do have an informative prior. Problems like Jaynes's burglar alarm.