Phase Margin (How far away you are from 180 Phase Shift) is a critical parameter used whenever designing any kind of feedback loop and testing for stability.
This is very to measure at the 0dB gain he pointed out, but lacked the phase diagram to show this shift.
Nonlinear systems responses to a sine signal are in general not just a change in phase and amplitude.
It works if the perturbation stays small and within a linearised version of the dynamics.
Here’s my attempt in a couple of sentences.
It takes time for the signal to propagate from input to output in any real circuit. If that time is a substantial fraction of the period under consideration then the input of the amplifier, which includes the feedback signal, cannot effect the output before it has moved. And if the delay through the amplifier is just wrong relative to the signal period one can end up in a dog chasing its own tail situation and the output oscillates.
The rest is just math. :)
P.S. this explanation also explains why we use phase and not seconds to measure the delay of the circuit. Because everything is relative to the input signal period and if we use phase we get that for free. No extra divide.