Yes, that's a good point. Tthat would be considered using a data augmentation prior (Sander Greenland has advocated such an approach).

You can dismiss this objection by replacing the coin with a novel experimental test with an easily computed expected probability of success– say, the very first test of spin-up vs. spin-down for silver atoms.

Frequentists can't claim relevant data sets for every experiment that has an obvious prior, without engaging in their own form of reference class tennis.