I studied math, so I can never understand the insistence that the truth of the fourier transform is that it converts from the time domain into the frequency domain. What if you apply the fourier transform twice? Something similar is done in voice processing, and it's perfectly valid. And the origin of the Fourier transform is not in the signal world, but rather in the diffusion world, and there's not necessarily a signal in time involved.

I think the concept of the basis of functions, a series of functions which you can combine to approximate any other function is something that engineers should be able to understand. Then you can see the time to frequency as a (very useful) application.

BTW, the visualization in the post as to how the transform works is awesome, and can also work with the function basis explanation instead of the frequency one. In fact, it might make even more sense! And the posterior mention of the cosine transform wouldn't need to be hand wavy about real and imaginary parts.

In any case, I've seen so many engineers insist in the time to frequency explanation that it must somehow be easier to understand for people. I just lament that the beauty of function spaces is lost in these explanations, as well as the underlying understanding of why Fourier transform is not only useful but deep.