The point of Gelman's reply is that the comic is actually comparing a Bayesian to an absurdly incompetent Frequentist, so there' really no conflict. No (modestly intelligent) Frequentist would mis-apply this methodology in this circumstance.
I've concluded that given a perfect, infinite-power MCMC simulator, I would always do a Gelman-style Bayesian analysis (with model falsification and improvement), but in practice, frequentist methods are computationally convenient.
Inference can be framed either way but means different things.
A Bayesian posterior P(H|D,M) is the probability that hypothesis H is true given data D and modelling assumptions M.
What does a frequentist p-value mean?
"Essentially, all models are wrong, but some are useful." — George Box