That's what I said, without using the word Taylor expansion, as this audience might not necessarily know it. I also explained in points 1 and 2 why many systems have can be modeled via small perturbations (i.e. Taylor approximation) around a stable equilibrium - they need to have low energy.

Also, just a reminder that it is the second-order term in the Taylor expansion that is relevant for harmonic oscillators. Zeroth-order term (constant) does not determine the dynamics. The first-order term (linear) is zero only for low energies, as any potential well at sufficiently low energies will be symmetric. The second-order term (quadratic) is what provides the restoring force towards the equilibrium.