The theory tells you the probability distribution of outcomes, which you'll see if you repeat the experiment prepared the same way a bunch of times (frequentism!), and that's all there is to it.

Well, the quantum probabilities are certainly frequentist. However, I don't suppose strict Bayesians deny that there are probabilities with frequentist interpretation. I am also not sure about the label ensemble interpretation. It seems that its proponents somehow deny the validity of QM for small, non-ensemblish systems, which is a position I don't subscribe to. After all, both collapse and many-world interpretations are no more Bayesian and no less frequentist than the ensemble one. The hidden parameters are deterministic, but have their own well known problems.

As for the frequentist-Bayes controversy, although I am probably more than you sympathetic to the Bayesian position, I have some sympathy for frequentism. I think both interpretation can coexist, with different sensible meanings of "probability".