they usually go about it with the Von Neumann-Morgenstern axioms, modified to refer to quantum mechanics: ...

Where did you get these axioms? They don't seem to make much sense, and they are in fact even mathematically inconsistent with QM (you have to at least replace (1-X) and X with sqrt(1-X) and sqrt(X) or you get non-unit state vectors).

I think it makes more sense to consider Von Neumann-Morgenstern axioms over density operators, which can be interpreted as (equivalence classes of) probability distributions on pure states. You can always consider these states as parts of the pure states of a larger system with unobservable degrees of freedom ( quantum state purification ).