> Computing large powers of large numbers is enormously unstable and needs to be avoided under basically all circumstances, that is what makes dealing with those polynomials difficult and cautioning people against this is obvious correct

But we don’t compute large powers of large numbers? Chebyshev is on [-1, 1]. Your large powers go to zero. And your coefficients almost always decrease as the degree goes up. Then to top it off, you usually compute the sum of your terms in descending order, so that the biggest ones are added last.