Differential equations aren't limited to describing stable systems, though, and there are chaotic systems that are also in some sense stable.

Ordinary differential equations can describe any system with a finite number of state variables that change continuously (as opposed to instantaneously jumping from one state to another without going through states in between) and as a function of the system's current state (as opposed to nondeterministically or under the influence of the past or future or some kind of supernatural entity).

Partial differential equations extend this to systems with infinite numbers of variables as long as the variables are organized in the form of continuous "fields" whose behavior is locally determined in a certain sense—things like the temperature that Fourier was investigating, which has an infinite number of different values along the length of an iron rod, or density, or pressure, or voltage.

It turns out that a pretty large fraction of the phenomena we experience do behave this way. It might be tempting to claim that it's obvious that the universe works this way, but that's only because you've grown up with the idea and never seriously questioned it. Consider that it isn't obvious to anyone who believes in an afterlife, or to Stephen Wolfram (who thinks continuity may be an illusion), or to anyone who bets on the lottery or believes in astrology.

But it is at least an excellent approximation that covers all phenomena that can be predicted by classical physics and most of quantum mechanics as well.

As a result, the Fourier and Laplace transforms are extremely broadly applicable, at least with respect to the physical world. In an engineering curriculum, the class that focuses most intensively on these applications is usually given the grandiose title "Signals and Systems".

One amazing application of spectral theory I always harp on when this topic comes up is Chebfun[1]. Trefethen's Spectral Methods in Matlab is also wonderful.

[1] http://www.chebfun.org/

I agree broadly with what you say. I didn't have time to make a more comprehensive comment.