We use sines and cosines, but not only sines and cosines. There are many other interesting sets of functions. For example, polynomials, or derivatives of the gaussian functions (called Hermite functions).

Bonus point: sines and cosines are polynomials evaluated on the complex unit circle.

> Basically the Fourier basis diagonalizes the convolution operator.

It also diagonalizes the second derivative, which is the linear operator that governs many physical processes, like wave propagation and heat diffusion.