Whenever any of this comes up I remember that physics professor's speech on first day of quantum mechanics that got viral:
“Nobody understands quantum mechanics. The people who came up with it don't understand it. I will do my best so that by the end of this course you don't understand it either, and so you can got out to the world and spread our ignorance.”
Quantum mechanics is illogical and stuff that happens makes no sense but can be recrcreated through experimentation....as long as you don't look at it.
Because it's part of reality, a foundational part of it even, it's logical basically by definition. If it wasn't, it would just mean our concept of logic is flawed.
Beyond that, we have perfectly logical and sensible descriptions for what is happening in quantum physics, the problem is just that we have more than one and don't know which is right.
I'd say we understand quantum mechanics better than most things.
We know more about the behaviour of an electron than we know about the oceans, the Earth, the sun, the weather, the stock market, the human body, prime numbers, and so on.
Do you mean "why" as in "why did X cause Y" or as in "why are things the way they are"?
In the former case, quantum mechanics is our most precise theory for coupling causes and effects, predicting the outcome of experiments to an incredible degree.
In the latter case, do we really have a grasp of that for anything? Why is the gravitational constant the value that it is? Why is pi the ratio of a circle's circumference and it's diameter? Mostly we ultimately have to say that it is so because we can observe that it is so. For quantum mechanics it is the same.
Oceans: We know the basic mechanics of currents, tides, chemistry, where all that water came from in the first place, and while there are a few known-unknowns it doesn't seem like a paradigm-shifting discovery is likely. They mystery is mostly because it's huge and we just can't look through it very well, and that there's too many physical inputs to track them all so models are abstractions by necessity.
The same goes for most of your list (I will not speak to prime numbers, I am an Earth Sciences guy and bad at higher math) in that we may not have a perfect map but we know the shape of it and where the probable gaps in understanding are. So the "why" is questions like "why do waves happen" or "why does the sun look yellow" or "why do we have embryonic 'gills'" and we have pretty good answers you can drill pretty deep into.
Pushing at the edges of physics is, I think, where the situation is flipped. We have very good models for the behavior of light but questions like "why is there a limit to the speed of information and why does light go that fast" and "why does it behave as a wave and also a mass-less particle" don't seem to have satisfying answers or even a means to be answered. Admittedly physics beyond its applications to organic chemistry is outside my education (again, math) but I try to keep up.
Whenever this picture comes up I remember that it's wrong - both electrons on it have the same spin, one is just rotated 180°, but it says +½ for one and -½ for the other, is like a part of the joke?
All electrons have spin 1/2, that's a property of it being an electron. They have a spin vector (the arrow shown) and whether it is in the same direction or opposite direction to the magnetic field it's in determines where it is plus or minus.
Now you might think "but what if it is not entirely aligned with the field, then it wouldn't be 1/2", which is true, on aggregate for large numbers of electrons, but if you ever look at a single electron its spin will either be "up" or "down" never any other orientation.
This is the kind of thing people are referring to when they say "no one understands QM", we know it is the case, we can measure it and predict it, but it makes no fucking sense.
I think where the sense starts to fall through is in remembering that it's not a ball though. It's a wave, and the spin only in one direction when interacting with something else, otherwise it's akin to three parts spinning clockwise, and one part spinning counter-clockwise. 🫠
I think the trick was to establish that disagreeing with the Copenhagen Interpretation makes you one of these idiots who can't comprehend Quantum Mechanics. Idiots like... Albert Einstein? Or... Erwin Schrödinger? You know, real morons.