What RPM do most gyros rotate at?
Even with his cardboard disc, the precession demonstrated is 90 degrees - as you have already been told, phase lag is frequently not 90 degrees, with some very large variations depending on rotor design blade inertia etc etc - but a gyro is ALWAYS 90 degrees.
Does the angle of gyroscopic precession change with density altitude? Phase lag does change with that variable because of the change in aerodynamic damping - how does that fit with your theory?
The similarity between precession and phase lag is why it has been used as a simple explanation of behaviour of a rotor but many far cleverer people than you and I have said they are not the same and one is not an explanation for the other.
What is your explanation for why phase lag is frequently not 90 degrees when gyro precession is always 90 degrees?
If all gyros behave exactly the same way, from the cardboard disc in the video through a spinning bicycle wheel to a proper gyro with a number of degrees of freedom designed to do a specific job, why do helicopter rotors not all follow the same rigid ('scuse the pun) reactions if it really is precession?
AC makes a very valid point about rigidity in space - you can't cherry pick the facets of gyro behaviour - if a rotor obeys one, it should obey all - and it doesn't.