The physics result is actually probably a consequence of that.
At the end of the day the whole lesson of modern math is that its useful to view things from many perspectives.
I can think of lots of other places I'd use fourier analysis (at least qualitatively as with doing diffusion modeling in my head) but you're right that sinusoids are more physically "real" whereas being possible to represent in any basis set is more "valid" if that makes any sense.
Not quite sure what the right word is on this one, but I agree "real" kind of suggests real oscillators underlying the phenomena. Square waves are less physical because of discontinuities in both the signal and derivative; nature really doesn't care for discontinuities.
The Gibbs phenomenon, for example, falls out naturally from the IFT of a frequency response where all the frequencies above some cutoff are zero.
I'm curious how the square wave frequency domain would describe the Gibbs phenomenon -- I think you'd have harmonics of the fundamental square frequency showing up as if the system were nonlinear.