But when one reaches the end, one finds merely the truism “It is this way because that’s what you get when you solve the equations.”
(yes, I have spent a lot of time working with physicists.)
In order for electrons to achieve stability (such as within an atom), they must create a standing wave. Similar to the notes on a guitar string, each electron orbital is simply a harmonic.
The complication comes from the fact that atoms are 3d, so instead you need to use spherical harmonics, but it's the same principle, just standing waves in 3d space. These should look familiar if you've seen rendered images of electron clouds: https://en.wikipedia.org/wiki/Spherical_harmonics
It is quite surprising that this intuition is basically never taught in classrooms.
We knew electrons had wavelike properties, but what equation did they obey? Even with hindsight it's still a difficult thing to derive - the relativistic answer is actually wrong, for example.
You reach the truism because the equations take years to learn - in engineering you can just take the result and run but in physics you have to be careful to understand what you're doing.
The Schrodinger equation for hydrogen isn't that difficult to solve (try it), but conveying that process in English is difficult.