I understand how to compute probability distributions and expected values and such from quantum states, but a lot of treatments of QM make it look like this is what the wavefunction is essentially for. To me this seems similar to assuming that energy exists to describe temperature: taking a large-scale emergent phenomenon as central simply because we were aware of it first. What aspect of quantum mechanics makes it natural to say that probability and 'non-determinism' should take a central role?
In particular I don't see any contradictions or ontologically unusual things that arise from treating a wavefunction as a real, deterministically evolving matter wave, with collapse arising from considerations of large-scale quantum statistics (a subject with which I am admittedly unfamiliar). Probability is simply a matter of ignorance (i.e. of the underlying quantum state of the environment/measuring device), as usual. Uncertainty principles are an obvious requirement for any wave theory, and non-commuting observables simply arise from this and the fact that we can no longer pretend to measure a system and leave it intact. If this is correct then I have absolutely no idea why people are confused/conflicted about quantum foundations, which is why I suspect I'm missing something. If simple wave effects are what 'non-realism' (and the lack of definite position, momentum etc.) amounts to, I see no reason why anyone would be even remotely uncomfortable with it.
Now I don't want the question to seem vague/philosophical, so I'd like to point out that I'm interested in experimental results (gedanken or otherwise) or uses of/proofs in QM that make the assumption that wavefunctions are really about probability natural, rather than 'probability density as modulus' just being a feature that wavefunctions happen to have.