Independently reproducing Jaynes's analysis is excellent, but calling him "cheeky" for "implicitly us[ing] different estimators" is not fair given that he's explicit on this point.

I was wrong to say that Jaynes implicitly used different estimators for the two methods. After the example he does mention it, a fact I missed due to skipping most of the post-mortem. I'll edit my post higher up to fix that error. (That said, at the risk of being pedantic, I did take care to avoid calling Jaynes-the-person cheeky. I called his example cheeky, as well as his comparison of the frequentist CI to the Bayesian CI, kinda.)

It's a frequentism-v.-Bayesian thing to the extent that correct coverage is considered a sufficient condition for good frequentist statistical inference. This is the fallacy that you rolled your eyes at; the room full of shocked frequentists shows that it wasn't a strawman at the time. [ETA: This isn't quite right. The "v.-Bayesian" part comes in when correct coverage is considered a necessary condition, not a sufficient condition.]

When I read Jaynes's fallacy claim, I didn't interpret it as saying that treating coverage as necessary/sufficient was fallacious; I read it as arguing that the use of confidence intervals in general was fallacious. That was made me roll my eyes. [Edit to clarify: that is, I was rolling my eyes at what I felt was a strawman, but a different one to the one you have in mind.] Having read his post-mortem fully and your reply, I think my initial, eye-roll-inducing interpretation was incorrect, though it was reasonable on first read-through given the context in which the "fallacy" statement appeared.