If there is, I'm not aware of it. Orthogonal polynomials come up in random matrix theory. Maybe there's something there?
But your last paragraph is exactly it... it is a "basic" fact but the consequences are profound.
Could you guys recommend an easy book on this topic?
Which topic?
Some sort of numerical analysis book that covers these topics - minimax approx, quadrature etc. I’ve read on these separately but am curious what other sorts of things would be covered in courses including that.
I would check out "An Introduction to Numerical Analysis" by Suli and Mayers or "Approximation Theory and Approximation Practice" by Trefethen. The former covers all the major intro numerical analysis topics in a format that is suitable for someone with ~undergrad math or engineering backgrounds. The latter goes deep into Chebyshev approximation (and some related topics). It is also very accessible but is much more specialized.
But I guess what I was asking was: is there some kind of abstract argument why the semicircle distribution would be appropriate in this context?
For example, you have abstract arguments like the central limit theorem that explain (in some loose sense) why the normal distribution is everywhere.