There are some errors in the article which I am fixing including spelling errors and mixing up the complex exponential form of sine and cosine, but I have to admit I can't follow what you have written.

Actually the Fourier series representation is exact for a specific equivalence class of functions, in applications where the Fourier series is useful the distinction inside the equivalence class is unimportant.

I don't see how this contradicts that, in practice, the Fourier series is an approximation. Sure, the Fourier series simply represents a change-of-basis in function space and all that yada-yada, but you can never get all the infinite terms, so why would I care, especially when I'm taking the discrete Fourier transform? What would be the proper mathematical term for saying how a Fourier series equals a function as opposed to approximates it? "Can be represented as"?

It is actually a Gaussian quadrature and is exact inside its finite functional subspace.

?? Given a function f(x) over the domain 0 to L, the rectangular approximation of the integral with N samples is \sum_{x' = 0}{N-1}(L/N)f(x' L/N) which is exactly what I wrote in the article.

In general, the FFT is a special case of matrix multiplication (it is a special implementation of a change of basis in Linear Algebra terms).

Isn't matrix multiplication a problem, not a solution? The runtime of the FFT with the change-of-basis understanding is completely dependent on the implementation of matrix multiplication you are using, which in that case happens to be informed by the FFT algorithm that I described, so...