IMO, the perceived inscrutability of quantum mechanics is mostly the result of the dominance of the "shut up and calculate" approach to the question of interpretation.

Don't get me wrong, I understand why focusing on the math and ignoring the rest is a good approach to have when you're actually doing practical scientific work in the field. But if you're wanting quantum physics to "make sense", then you're asking about what it actually says about the underlying reality, and that's a question that requires thinking about interpretations of QM.

Once you start thinking about it in terms of a particular interpretation (or category of interpretation), then QM makes as much sense as any other high level physical theory (certainly no worse than general relativity.)