I mean, the biggest assumptions that most influence the inferences one makes are rarely "statistical" in the sense that they can actually be incorporated in a p

I mean, the biggest assumptions that most influence the inferences one makes are rarely "statistical" in the sense that they can actually be incorporated in a particular analysis via a prior. They tend to be structural assumptions that represent some fundamental limit to your current state of knowledge, no? Certainly this is domain-specific, though. I once read a Gelman blog post or paper that argued Frequentists should be more Frequentist (i.e., repeat experiments more often than they currently do) and Bayesians should be more Bayesian (i.e., be more willing to use informative priors and or make probability statements beyond 95% credible intervals). Or something like that, as I am paraphrasing. That always seemed reasonable. Either way, the dueling--and highly simplified--caricatures of Bayesians vs. Frequentists vs. likelihood folks is largely silly to me. Use the tool that works best for the job at hand, and if you can answer a problem effectively with a well designed experiment and a t-test so be it.