I'm intrigued by your quote "all physics is either harmonic oscillators or phase transitions". Could you please elaborate on that? Just looked up online sources but can't seem to find this quote anywhere.
Obviously this is not exhaustive. But it gets you further than you naively think it should. Let's look at the harmonic oscillator, why is the thing probably a harmonic oscillator? Well the potential is typically an analytic function, and we are near a resting point, so the linear order in the Taylor expansion vanishes and the potential is approximately V(x) ~ x^2. Harmonic oscillator.
Say you want to look at a quantum field theory. Well actually defining one is hard. The only case where we know how to is for a free field theory. And a free field theory is actually a collection of harmonic oscillators.
Now interacting QFT, which underlies all of observed matter, is built by gluing together harmonic oscillators in a clever way. You know Feynman diagrams? The lines in a Feynman diagram represent the particle behaving as if it was made up of independent harmonic oscillators (free field). At the vertices we just bump the harmonic oscillators around a bit. So standard QFT is a very clever shuffling around of harmonic oscillators (lots of group representation theory organises the shuffling, and several Nobel prizes worth of physics are contained in the details).
Obviously there is plenty of physics that does not fit into either of these paradigms (GR, non-linear dynamical systems, atomic and molecular physics). But they both are utterly fundamental and enormously powerful tools in two very prominent branches in physics: High energy and condensed matter.