In contrast, a few simple desiderata for "logical reasoning under uncertainty" establish probability theory as the only consistent way to do so that satisfy those criteria. Sure, other criteria may suggest some other way of doing so, but no one has put forward any such reasonable way.

Could Dempster-Shafer theory count? I haven't seen anyone do a Cox-style derivation of it, but I would guess there's something analogous in Shafer's original book.