I believe the author assumes that it's clear to the reader that there is a distinction between how a mathematical object is defined, and how it's computationally used. A polynomial can be defined as a power series, but it's not how they are computationally used. In this sense, the author was mistaken.
But it's not that the problems are "circumvented", in the sense that it's a kind of a hack or a patch, but they are solved, in the sense that there is a systematic way to correctly compute with polynomials.
Actually they aren't. You never compute high powers of the argument when working with specialized bases.
You use the recursive formula that both the Bernstein basis and the orthogonal polynomial bases are endowed with. This is implemented in numpy, so you don't have to do anything yourself. Just call, for example, np.polynomial.legendre.legvander to get the features for the Legendre basis.
And a basis orthogonal over [-1,1] is easily made orthogonal over arbitrary interval. Take p_i to be the i-th legendre polynomial, then the basis composed of
q_i(x)=p_i(2(x-a)/(b-a)-1)
is orthogonal over [a,b]. Each q_i is itself a polynomial of degree i, but you never use its coefficients explicitly.
There is an entire library for computing with polynomial apptoximants of functions over arbitrary intervala using orthogonal polynomials - Chebfun. The entire scientific and spectral differential equations community knows there are no numerical issues working with high degree polynomials over arbitrary intervals.
But it's not that the problems are "circumvented", in the sense that it's a kind of a hack or a patch, but they are solved, in the sense that there is a systematic way to correctly compute with polynomials.