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At the end of the day, analytical Fourier analysis doesn't get nearly as much practical use as numerical approaches, and you don't really need THAT much background to make sense of DFTs.

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.

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Analytical fourier analysis is needed for many other fields of math though. For example in statistics it's very central for dealing with density functions. I'd guess in many fields of physics it's also very crucial.

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.

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