As far as I can understand you, you seem to think that because humans aren’t always rational in the sense described by Cox’s postulates, probability theory only “models” human reasoning under uncertainty. You also seem to think that probability theory is just “squiggles on paper”.

Only models? Just squiggles on paper?

You've misunderstood the article, I think. Probability theory (the Kolmogorov Axioms) does model correct degrees of belief and describes normatively what they should be. It also models "long-term frequencies" in the sense that the Kolmogorov Axioms also apply to such things.

None of this requires the word "probability" to refer to degrees of belief. You don't even need a word at all to do the math and get the right answer. It's convenient to use the word that way though, since we already have a word "frequency" that refers to the stupider idea.

(And also I suspect that most people learned the word at school mostly by being given examples of likely and unlikely things. For them, "probability" refers to the little progress bar in their mind that goes up for more likely things and down for less likely things [ie. degrees of belief]. And thus many frequentists may commit philosophical errors when they try to define it as frequencies then use the intuitive definition to draw a conclusion in the same argument. This alone is a good reason to use "probability" for beliefs and "frequencies" for, well, frequencies.)