The proof in Jaynes applies proves that if you want to assign plausibilities to propositions, and those plausibilities are going to be real numbers, and P(a^b|c) is a function of P(a|b^c) and p(b|c) and P(not a) is a function of P(a) and all those functions are twice-differentiable, and P satisfies a certain density requirement, then P has to be isomorphic to probability.

It just doesn't have the same philosophical punch as "a few criteria that just seem like common sense that everyone should agree are desirable in a reasoning system." when you actually spell out the assumptions and they contain seemingly unjustified technical things like differentiability and density.

There are a bunch of rigorous variations on Cox's theorem, but as far as i know there is nothing that lives up to the hype.