For example the current version of the Fourier analysis article on Wikipedia says the study is:
[…] named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.
And as mentioned in e.g. Origin of the Fourier transform (1878), the book often cited re. Fourier series has a title that translates to Analytical Theory of Heat.
But I don't see any sines or cosines in the (rather simple-looking) heat equation.
Nowadays it seems that techniques based on Fourier's work are particularly famous through their use in Digital Signal Processing, in which FFTs/DFTs are often applied to data series that are directly related to sine waves. When I think of Fourier transforms I think of analyzing data that's full of signals and noise and rapid changes in levels, not something like a hot spot slowly averaging itself out through a chunk of metal.
There's another related question here, What are some good references elucidating the discovery/creation of Fourier Series?, but as that asked for references the answers there don't focus on specifically answering this question per se, and in some ways could be considered link-only answers relative to this question.