True, the Fourier transform can be viewed as a specialization of wavelet transforms, so in that sense wavelets are fundamental.
Like many mathematical subjects the specialization is treated before some historical generalization in educational contexts (if the generalization is ever discussed, as it doesn't seem to be here).
Practically speaking, while wavelet theory is fascinating, you can get plenty of DSP done without knowing it; but not having some understanding of the DFT is crippling.