You can’t physically construct a speaker that makes a sawtooth wave. Its cone would need to change velocities from -n to +n or vice-versa instantaneously in order to generate the ‘teeth’ of your wave. The air particles you are moving would likewise need to instantly accelerate. That is a physical impossibility - these things have mass, accelerating them requires force, infinite acceleration requires infinite force.

Those physical constraints manifest as limits on the frequency of sinusoidal harmonics it is possible for you to put into the wave; for the medium to carry; and for you to physically detect at the other end.

Mathematicians don’t break functions down into sinusoidal harmonics because they like trig functions. They do it because they fundamentally are what’s happening.