Very cool explanation. Something I've come across a few times on here was wanting to explain how 12tet "includes" or handles approximations of intervals from other scales, and how that affects the musical choices of musicians or especially the notation choices for transcription of improvised music.

But it's impossible to explain without getting into like, what is even the problem solved by tuning systems. Without the intuition that comes from making music, programmers and engineers see the fractions & obvious series and get too fixated on finding the "perfect" system. When these are much more physical tools, created over time to make certain processes easier. Tuning systems are more like a woodworker's knives than like the unit circle: being perfect does not make them better tools for creation if they are already fit enough.

12tet is a practical compromise between overtone approximation and playability.

Any fretted or keyed acoustic instrument with more than 12 notes/octave is extremely difficult and expensive to build and hard to tune and play.

The music is also hard to notate, because there are so many more possible pitch positions.

So 12 notes became a practical default for instrument builders. And from there, ET became a practical default for tuning to allow smooth-ish modulation through different keys. The errors in the tuning are the same for every key, so the tuning relationships stay the same while the key root changes.

That doesn't quite happen because overtone perception and beating effects aren't completely linear. But it avoids the obvious out-of-tune notes you get when playing in distant keys with non-ET tunings.