One deborah mayo (no link) had this to say:
p-values are NOT likelihoods, however, they permit computations that Bayesian likelihoods alone cannot. They allow evaluating the probability that the testing procedure would have resulted in a less impressive departure (from the null) under the assumption the null is true, and also under the assumption of varying discrepancies from the null. It's a small part of the panoply of methods that use error probabilities. Guess what? Bayesians are the ones who only use likelihoods conditional on the observed value! So no error probabilistic assessments are possible. Oh, but there's a prior you say? No error control there either--just what someone believes, and very few scientists want to mix their prior beliefs into the study. The point of the research is to test claims--not beg the question by imputing prior beliefs!
So, yes, once again those villagers out to slay the p-value Monster begat by Dr. Frankenstein turn to the turgid ravings of an ancient Presbyterian minister. Oy vey!!