Hi Matt,
Thank you so much for your interesting point. I would like to focus on the case of quantum theory. As you pointed out, we can consider the single-case probability. On applying the single-case probability to the single event in quantum mechanics, I think that we need the same (i.i.d.) samples to verify this probability. The statement of i.i.d. samples implies the rule of probability theory. This seems to be logical mistake. Also, I think that the fact that the operational derivation of physical systems depends on the interpretation of probability seems to be strange.
Yutaka