You can do better than frequentist approach without using the "magic" universal prior. You can just use a prior that represents initial ignorance of the frequency at which the machine produces head-biased and tail-biased coins. (dP(f) = 1df). If you want to look for repeating patterns, you can assign probability (1/2)(1/2^n) to the theory that the machine produces each type of coin on a frequency depending on the last n coins it produced. This requires treating a probability as a strength of belief, and not the frequency of anything, which is what (as I understand it) frequentists are not willing to do.

Note the universal prior, if you can pull it off, is still better than what I described. The repeating pattern seeking prior will not notice, for example, if the machine makes head biased coins on prime-numbered trials, but tailbiased coins on composite-numbered trials. This is because it implicitly assigns probability 0 to that type of machine, which takes infinite evidence to update.