[kgwgk]

The calculation Input -> [Model] -> Output is not wrong.

I don't know what are you trying to say but it's no longer related to whether Bayesian methods and frequentist methods give the same answers. They don't even try to represent the same questions.

PS: I didn't have time to reply to your later comment, so here is a final comment reaching for common ground.

> frequentist conceptual machinery

Just to be clear, that's not what I've been talking about. As I'm tried to make clear several times I'm talking about what is commonly understood as frequentist methods. I gave the concrete example of confidence intervals. I started the conversation with a explicit question: What do you call "frequentist methods"?

I agree that if we define "frequentist conceptual machinery" to be "probability" we can do Bayes with it.

[whatshisface]

Frequentism and bayesianism are more than separate collections of formulas, they're philosophical stances on interpreting theorems about statistics. That's where the separate language comes from. It might not be taught very often (because it's inconvenient and wordy), but it is actually possible to address anything in frequentist language, even cases where you're combining separate sources of differently-weighted evidence. The same goes for bayesian language.