Thanks for that alternate way of thinking about it. As an engineer, half of my intuition about the FFT comes from the notion of the "frequency domain", and the other half from "change of basis".
The explanation you gave, of FFT as an alternative representation of a polynomial through its value at selected points, gives that nice intuition about pointwise multiplication, without appealing (directly) to the "convolution-turns-into-multiplication" catchphrase.
The explanation you gave, of FFT as an alternative representation of a polynomial through its value at selected points, gives that nice intuition about pointwise multiplication, without appealing (directly) to the "convolution-turns-into-multiplication" catchphrase.