This illustrates the idea that, from a Bayesian perspective, implausible claims require more evidence than plausible claims. Frequentists have no formal way of including that type of prior information.

Yeah, the best they can do is require a stricter significance level. (IIRC, the justification for the standard "statistical significance" test involves something like taking the position of maximum ignorance by assuming that the null hypothesis is 50% likely to be true and then applying Bayes's Theorem to get a posterior probability...)