> Because an FFT (short for "Fast Fourier Transform") is nothing more than a curve-fit of sines and cosines to some given data

That is not even wrong. A Fourier transform is a basis expansion. In particular, the full expansion is exact (not just an approximation). Of course, truncated expansions are approximations.

The actually interesting part: Why is this basis expansion so much more useful than, e.g. expanding into some eigenfunctions, Hermite polynomials, etc.? The decomposition into (complex) exponentials converts between addition and multiplication, i. e. sin(x+y), cos(x+y) you get from multiplying sin(x), cos(x), sin(y) and cos(y). This in turn has important implications such as turning derivatives into multipliers. More generally you can consider nonlinear Fourier transforms with different groups and generators other than exponentials.

TLDR: It is a transform. What you are transforming between is what makes it so useful.