You need analysis to make sense of infinite series, indefinite integrals, exponential functions, etc.
I know that, for myself, revisiting Fourier Analysis after going through DFTs in my numerical analysis classes made a lot more sense, and I kinda wish I had started with that angle in the first place.
But for a lot of fields you just need the DFT and the calculus stuff can be mostly a distraction. How I finally figured out FT is something like the infinitesimal limit of DFT.
The calculus course is way too bloated for historical reasons that haven't mattered for more than a century. Pushing everything into that context only serves to make the contents hard to understand and seemingly useless.