I've read that we can typically reconstruct any complex wave form from other periodic waves. So it doesn't seem like sines are something special.

But apparently, mass on a spring behaves in a sinusoidal motion https://youtu.be/n2y7n6jw5d0?t=968

And sines are also an approximation of harmonic motion of a pendulum. https://youtu.be/p_di4Zn4wz4?t=370

Many physical systems resonate or oscillate produce quasi-sinusoidal motion.

https://ccrma.stanford.edu/~jos/mdft/Why_Sinusoids_Important...