The underlying theory that justifies most inference (Bayesian, minimax, etc.) is decision theory, which is a subset of the theory of games. Savage’s book on the foundations of statistics has a very nice discussion of why this should be. I learned it from Kiefer’s book, which is the only book I know of that starts there. Lehmann or Casella both get to it later in their books.

The justification for p-value is actually the Neyman-Pearson theory of hypothesis testing. The p-value is the critical value of alpha in that framework. I wrote a couple of expository articles for clinicians going through this if you’re interested.

Jaynes was a wonderful thinker, but be aware that a lot of the rational actor theory breaks down when you don’t have a single utility function. That is true of using classes of prior (see the material towards the end of Berger), or in sequential decision problems (look at prospect theory in psychology, where the overall strategy may have a single utility function, but local decisions along the way can’t be described with one). So the claims in the middle of the 20th century for naturalness of Bayesian reasoning haven’t held up well.