It’s more that wave functions are ubiquitous in all areas of physics. It generally falls out of the fact that we can model tons of stuff with second order differential equations, which often result in models that have oscillator/wave-like behaviors

You are mixing up two different (but related) terms. The wave equation is a second-order partial differential equation which has some solutions in the form of waves. A QM wave function is a particular case of a wave equation, yielding a probability amplitude.

One significant application of the wave equation in pre-quantum physics occurred when Maxwell was able to derive, from the electromagnetic theory he was developing, a wave equation having solutions in which electromagnetic waves propagate at the speed of light.