Like, suppose someone hands me a biased coin. A frequentist can flip that coin 10 times, then make a bound on the true bias of the coin that holds with the promised degree of confidence. (E.g. they can make a bound that will hold on 95% of such experiments.)

Bayesians can't do this. There are lots of other advantages to the Bayesian approach, but to frequentists, nothing could be worth giving up "coverage" (confidence intervals obeying their guarantees).

P.S. When looking at those ten coin flips, and trying to estimate the true bias, the frequentist and bayesian will have the following argument:

Bayesian: The probability that the true bias is b is __some formula___

Frequentist: Are you insane? The true bias of the coin is what it is! It's a number! Just because you don't know what that number is doesn't mean anything probabilistic is going on.

Bayesian: Well it is extremely useful to be allowed to make such statements.