I'm sure they would! But there's a pretty strong argument it's a step in the right direction. Actually, this link makes a more modest proposal that papers should report likelihoods rather than p-values. This avoids reported results results depending on priors (which perhaps we don't trust authors to choose well), though a reader can easily impose themselves if they want to.
Probably.. but with Bayesian methods at your model/data updates its priors, rather than you effectively embedding your prior beliefs into your models via selectively choosing tests that support them.
In my view, "prior" may be a misnomer. There is nothing that I'm aware of in Bayes' theorem to suggest that you have to formulate your priors before gathering or analyzing your data. I would describe priors as constraints that are included in an analysis, to narrow the results based on additional information that you're aware of. Bayes' theorem mainly provides a framework for computing what happens when you do that.
https://arbital.com/p/likelihoods_not_pvalues/
You still have the option to muck with things by choosing your hypothesis class in a bad way-- nothing can really replace publishing data!