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.
I'd suggest: Trefethen, Lloyd N., Approximation theory and approximation practice (Extended edition), SIAM, Philadelphia, PA (2020), ISBN 978-1-611975-93-2.
But your last paragraph is exactly it... it is a "basic" fact but the consequences are profound.