Even if the claim is worded like that, it implies (incorrectly) that correct reasoning should not involve steps based on opaque processes that we are unable to formulate explicitly in Bayesian terms.

You misunderstand. There was no normative implication intended about explicit formulation. My claim is much weaker than you think (but also abstract enough that it may be difficult to understand how weak it is). I simply assert that Bayesian updating is a mathematical definition of what "inference" means, in the abstract. This does not say anything about the details of how humans process information, and nor does it say anything about how mathematically explicit we "should" be about our reasoning in order for it to be valid. You concede everything you need to in order to agree with me when you write:

You could [justify intuitive judgements in Bayesian terms] if you had a way of reverse-engineering the relevant algorithms implemented by your brain,

In fact, this actually concedes more than necessary -- because it could turn out that these algorithms are only approximately Bayesian, and my claim about Bayesianism as the ideal abstract standard would still hold (as indeed implied by the phrase "approximately Bayesian").

Of course, this does in my view have the implication that it is appropriate for people who understand Bayesian language to use it when discussing their beliefs, especially in the context of a disagreement or other situation where one person's doesn't understand the other's thought process. I suspect this is the real point of controversy here (cf. our previous arguments about using numerical probabilities).