If you don’t have any notion of how likely the coin is to be biased or how it might be biased then you just can’t do the analysis at all.

You can certainly do the frequentist analysis without any regard to the distribution of coins from which your coin was sampled. I’m not well studied on this stuff, but I believe the typical frequentist calculation would give the same results as the typical Bayesian analysis with a uniform prior distribution on “probability of each flip being heads.”