Fourier's fundamental discovery was that the most natural building blocks for periodic functions are complex exponentials. These in turn were based on Euler's identity, linking complex exponentials to sines and cosines, and which is algebraic (easy to differentiate, integrate, manipulate) and also encodes rotations and waves. So, a good reason to study complex analysis and include it in the engineering maths program.

This setup then allowed Fourier to take the limit of the Fourier series using a complex exponential expression. But keep in mind, this is all in the context of 19th century determinisitic thinking(see Fourier's analysis of the heat equation 1807, and Maxwell's treatise on 'Theory of Heat' c. 1870s), and it ran into real-world limits in the late 19th century, first with Poincare and later with sensitive dependence on initial conditions. . Poincare showed that just because you have a deterministic system, you don't automatically get predictability. Regardless this Fourier transform mathematical approach worked well in astronomy (at least at solar-system scale) because the underlying system really was at least quasiperiodic - essentially this led to prediction of new planets like Neptune.

But what if you apply Fourier transform analytics to data that is essentially chaotic? This applies to certain aspect of climate science too, eg, efforts to predict the next big El Nino based on the historical record - since the underlying system is significantly chaotic, not strictly harmonic, prediction is poor (tides in contrast are predictable as they are mostly harmonic). How to treat such systems is an ongoing question, but Fourier transforms aren't abandoned, more like modified.

Also, the time-energy quantum mechanics relation is interesting though not really a pure QM uncertainty principle, more like a classical example of a Fourier bandwidth relation, squeeze it on one axis and it spreads out on the other - a nice quote on this is "nothing that lives only for a while can be monochromatic in energy." Which sort of leads on to virtual particles and quantum tunneling. (which places a size limit of about 1 nm on chip circuitry).

The bottom line is that if you're applying elegant and complex mathematical treatments to real-world physical problems, don't forget that nature doesn't necessarily follow in the footsteps of your mathemetical model.