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What if "Bayesian rationality" were deleted?
How did "Bayesian rationality" get discovered, except by the usual practices of scientists? (I won't say "the scientific method", partly because it's really fuzzy and so the "the" at the beginning of the phrase is deceptively concrete, and partly because I don't think that the process is as tidy as descriptions of the scientific method make it out to be.)
If we're looking for an error-correcting system, we need to look for a vast number of weak epistemological principles, on the level of "if event X is followed almost immediately by event Y, guess that X is generally followed almost immediately by Y", along with perceptual details of "how long?" and "what should count as an event?".
They would probably be fiercely embodied, but that's not actually a problem - we're fiercely emphysicsed, after all.
I don't know about deducing the entire mindset & toolbox of 'Bayesian rationality,' but knowing Bayes' theorem is the key part of it, and I wouldn't expect that to be too hard to reconstruct if you knew what you look to for.
Bayes' theorem follows trivially from the definition of conditional probability, and that definition is itself quite intuitive. So in theory, once you have a feel for what probability is, it'd be quite possible to get to Bayes' theorem. I haven't read Huygens' 1657 book on probability theory, but if it was any good, I bet Huygens knew enough about it to beat Bayes to Bayes' theorem by a century.
Chapters 1 and 2 of Jaynes' Probability Theory: The Logic of Science show how Bayes' theorem follows necessarily from certain basic principles of plausible reasoning. In some sense all roads lead to Bayes when trying to derive a consistent mathematical procedure for manipulating degrees of plausibility.
You are quite right. I thought about mentioning the Cox-Jaynes road to Bayes' theorem in my post, but decided that someone trying to reconstruct Bayes' theorem would be more likely to get to it by muddling through intuitively via conditional probability.