What you're on to is that all of the big popular examples of "quantum weirdness" (uncertianty, entanglement, tunneling) emerge from a very small and elegant set of axiomatic postulates. That's the beauty of physics.

One such group of postulates that chemists like to use can be found at http://vergil.chemistry.gatech.edu/notes/quantrev/node20.htm... . You can probably beat the elegance of those if you're willing to step further from what's actually used in calculations, but that's not what chemists like to do.

Here's a better shot at being axiomatic and elegant, but it's a lot less clear (from a lay-perspective) how they relate to reality: https://en.wikipedia.org/wiki/Dirac%E2%80%93von_Neumann_axio...

I find Lucien Hardy's discussion enlightening:

Quantum Theory From Five Reasonable Axioms

https://arxiv.org/abs/quant-ph/0101012