The author of the video was simply pointing out that there exists a secondary oscillatory phenonema not unlike a gyroscope under-going precession. As to whether precession is actually defined AS a simple harmonic oscillator I am not sure about.In the case of a pool ball, what is the observation that best illustrates gyroscopic effect?
If you are still curious here is the mathemaics behind it. As you can see the equations look similar to gyroscopic precession (Or do you want to bring up the difference between Einsteinian and Newtonian theories?)
This is a fun video, don't get caught up on semantic details!