What you have labelled Cox's theorem is not Cox's theorem; it's some version of the Dutch book argument and/or Savage's theorem. Cox's theorem is more like this. (I am the author of those blog posts.)

Also, you've misstated the complete class theorem -- it's actually weaker than you claim. It says that every estimator is either risk-equivalent to some Bayesian minimum posterior expected loss estimator (BMPELE) or has worse risk in at least one possible world. Conversely, no non-BMPELE risk-dominates any BMPELE. (Here "risk" is statistical jargon for expected loss, where the expectation is taken with respect to the sampling distribution.)