No.

This paragraph has nothing to do with numerics. It is about the fact that continuous functions can not be approximated globally by polynomials. So you need to restrict to intervals for reasons of mathematical theory. This is totally unrelated to the numerical issues, which are nowhere even acknowledged.

But what's the point in acknowledging numerical issues outside of [-1,1] if polynomials do not even work there, as author explicitly notes?

All polynomials "work" globally. That some polynomials form an orthonormal basis over certain intervals is essentially irrelevant.

The author does not address the single most important reason why high degrees of polynomials are dangerous. Which is pretty insane to be honest, obviously to be honest you have to at least mention why people tell you to be cautious about high degree polynomials AND point out why your examples circumvent the problem. Anything else is extremely dishonest and misleading.

Is there a mathematical proof or basis to back what you’re saying?

That polynomials do not approximate continuous functions globally??

That is pretty obvious. Consider that every polynomial grows arbitrarily large, but not every continuous function does.

There is a simple corollary to Stone-Weierstrass that extends to infinite intervals, but requires the use of rational functions.