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.”

I guess it depends on exactly what kind of information you want. Frequentist analysis will give you the probability of getting an exact 100/100 split in a world where the coin was fair. That probability is about 0.056. Or you can go for p values and say that it's well within the 95% confidence interval, or whatever value. But that's not very useful on its own. What we typically want is some notion of the probability that the coin is fair. This is often confused with the probability of getting the result given a fair coin (e.g. 5% probability X would happen by chance, therefore 95% probability that the null hypothesis is false) but it's very different. In this context, the question people are interested in is "how likely is it that Fleming/Mendel p-hacked their results, given the suspicious perfection of those results?" Analogous to "how likely is it that the coin is fair, given the exact even 100/100 split we got?" And for that, you need some notion of what unfairness might look like and what the prior probability was of getting an unfair coin.