seems circular - if you limit the world to small perturbations over a stable equilibrium of course you get simple harmonic motion, push a spring down a bit it pushes back and starts bouncing, the interesting insight is why can you limit the world to that?

i suppose it’s because most interesting forces are conservative, i.e. if motion doesn’t dissipate energy (like friction, unlike electromagnetism) then energy is conserved so perturbations must bounce. OP contains the key insight- dissipating systems dissipate proportional to total energy, so low energies are approximately stable, IIUC.