Here's the thing: the frequentist in the comic has made an error even by frequentist standards, and that error is equivalent, in Bayesian thinking, to choosing an inappropriate prior.
The problem is that many frequentist techniques implicitly choose a prior for you. That's handy since choosing an appropriate prior is hard. But it also abstracts away the choice of prior.
If a Bayesian makes this mistake, anyone can look at their math and say "There. That's where you chose a bad prior."
If a frequentist makes this mistake, you have to have a complicated analysis to explain why the method used is inappropriate.
I agree in with this analysis. The cartoon is, of course, a caricature but highlights the nature of the essential problem you point out, which is that unraveling the line of reasoning carved out by a frequentist is not a straightforward matter, even if the argument is correct. From the Bayesian perspective all arguments are deductive once you have all the information. A typical reply to this observation is that in most cases it will be clear how the steps of a calculation translate into the various assumptions needed to clarify the argument. Of course, what counts as "typical" depends on the kinds of questions one asks. If there are only a few kinds of problems in your field, then maybe you can get away with heuristic lines of reasoning, patching up problem in special cases as needed. But if you are, say, trying to write a general purpose software for statistical analysis it will not be possible to rely on inductive lines of reasoning particular to a certain field. This might partially explain the popularity of the Bayesian approach in machine learning circles.
The problem is that many frequentist techniques implicitly choose a prior for you. That's handy since choosing an appropriate prior is hard. But it also abstracts away the choice of prior.
If a Bayesian makes this mistake, anyone can look at their math and say "There. That's where you chose a bad prior."
If a frequentist makes this mistake, you have to have a complicated analysis to explain why the method used is inappropriate.