Nature is inherently probabilistic

It’s not really an “issue” that quantum mechanics is probabilistic — this is simply how nature works! Your view of “probabilistic being an approximation” is exactly the point of view which Einstein had about quantum mechanics — together with Podolsky and Rosen he published the most serioius criticism of QM in a seminal paper. It is sometimes also called “EPR paradox” after their initials. The wiki article about it is reasonably good. This paradox was famously taken out of the world with the experimental verifications of the violations of the so-called “Bell inequalities”. Because of them, we now know that nature is really inherently probabilistic and one can prove that it’s not just us not knowing the underlying “fundamental” theory which makes the world seem probabilistic.

Now, to your second point: The measurement process really is one of the darkest corners of quantum mechanics. There is probably not a single person who would say that they are happy with the “collapse of the wavefunction” picture we have right now. Unfortunately, no real alternative exists and it does match experiment incredibly well — and that is what counts.

Trying to make sense of quantum physics with the help of green tea.