I still don't understand your argument. Of course, the issues with set theory were mathematical issues and had to be resolved to have a solid foundation of set theory and given its importance as a basic tool in all branches of mathematics also of mathematics as a whole.
Quantum mechanics, at least the non-relativistic flavor, has no mathematical problems as far as I can see. This of course includes Born's rule, and given this set of postulates/axioms you have a special kind of probability theory, if you just consider QM as a mathematical construct.
Now when it comes to physics all you need in addition is a relation between what's observed in nature and the formalism, and with the minimal statistical interpretation there are no problems either. If there were problems, QM wouldn't be as successful as it is in describing the observations. If there is a discrepancy, we'd learn something fundamentally new. That's how (theoretical) physics works.
Of course it's desirable to have mathematical sound and solid formulations of physical theories, but it's a different issue to consider a mathematical theory, which does not necessarily need to describe nature but just has to be mathematically sound and solid. E.g., Newtonian mechanics is a mathematically well-defined theory providing rules to derive differential equations and some general and beautiful tools for their solution. Particularly the entire field of Lie-group and -algebra theory has been triggered by it. Nevertheless it's not sufficient to describe all observations and thus had to be modified (including the necessity of new descriptions of space and time in relativity theory and QM).
Another type of trouble is classical electrodynamics. As long as you deal with continuum mechanical models of charged matter the issue from the point of view of mathematics is quite the same as with Newtonian mechanics. Here we have a well-defined classical relativistic field theory providing PDEs and some nice tools to solve them. Nevertheless we know it's incomplete as soon as it comes to models for matter in terms of point particles. There's no satisfactory closed dynamics for fields and point particles. The best one could come up with for more than 100 years is the Landau-Lifshitz approximation. Of course it's known that classical point particles are a lousy model of matter on a microscopic level and one has to use QT, and in this case the best one could come up with is relativistic QFT, which has the known problems as a mathematical theory, which however can be overcome in the typical pragmatic manner physicists ignore mathematical problems by dealing with hand-waving arguments. For a mathematician there's of course still a lot to be desired, and maybe one day there's a more comprehensive mathematically sound and solid microscopic theory that describes all observations as well as relativistic QFT does as an "effective theory" today.
All these examples show that there's a strong overlap between math and theoretical physics, but the goals are partially different. As a physical theory QM is amazingly successful, and there are no foundational problems. You just have to accept probabilistic arguments as a fundamental feature of nature. Quibbles with this view are metaphysical and have nothing to do with either math or the natural sciences.