" < Jaynes quote > ... If Nature is one way, the likelihood of the data coming out the way we have seen will be one thing. If Nature is another way, the likelihood of the data coming out that way will be something else. But the likelihood of a given state of Nature producing the data we have seen, has nothing to do with the researcher's private intentions. So whatever our hypotheses about Nature, the likelihood ratio is the same, and the evidential impact is the same, and the posterior belief should be the same, between the two experiments. At least one of the two Old Style methods must discard relevant information - or simply do the wrong calculation - for the two methods to arrive at different answers."

This seems to be wrong.
EY makes a sort of dualistic distinction between "Nature" (with a capital "N") and the researcher's mental state. But what EY (and possibly Jaynes, though I can't tell from a short quote) is missing is that the researcher's mental state is part of Nature, and in particular is part of the stochastic processes that generate the data for these two different experimental settings. Therefore, any correct inference technique, frequentist or Bayesian, must treat the two scenarios differently.