The connection between Kolmogorov's and Cox's foundations and frequentist vs. Bayesian interpretations is not clear to me.

Kolmogorov's axiomatization treats events as elements of a powerset rather than as propositions. Cox's axioms give you belief-allocation among any finite number of quantifier-free propositions, which then lets you derive countable additivity quickly from there.

Whereas I kinda think that distributions and degrees of belief ought to be themselves foundational, degrading to classical logic in the limit as we approach certainty. This is the general approach taken in the probabilistic programming community.