Hmm... I suppose it can come across as pedantic, but physical processes as a whole can do anything that can be done, hence it shouldn't come as a surprise that there exists a physical process that can perform a Fourier transform, or any computable process whatsoever.

I agree. To me it's more interesting that there exists a physical process that performs a Fourier transform than that there could exist such a process. If it can be done on pen and paper, it's already proven that it can be done in a physical system, since the pen and paper are physical.