It is mathematically possible to express a given signal as a sum of functions other than sines and cosines. With that in mind, why does signal processing always revolve around breaking down the signal into component sine waves?
You are not required to. Functions can be decomposed into a wide array of orthogonal basis functions, including the Bessel functions (in the Hankel transform) and the Legendre functions. The sine function just happens to be the overall simplest to deal with in the general case.
You refer to Fourier Series. The brilliance of Fourier was to use sin to express a function.You know that you can create any vector from the sum of some unit vectors.Exactly the same think happens here. The number you multiply the unit vectors is the coefficients in F.S. To answer to your question of why we use sin and cos is that they have (mathematicly) the same property like the unit vectors.In any book, you see (but you don't objerve),that those unit vectors are orthogonal to each other.Same thing with sin and cos.
A very good video on F.S is this: https://www.youtube.com/watch?v=le_gMPJFyJ8
Go to 11:50 to see what i mean with the term orthogonal.