IMO, flywheel drag controls only the rates of the 3 observed rotations, not the basic dynamic.
I believe that the flywheel drag produces the precession. No air drag, no precession, in that setting.
I think that without those fins the air moved from the flywheel would be quite less, so the difference of behaviour would come up, (if you could find a smooth flywheel).
I realized that I can replicate the experiment with my gyroscope; there are some differences but they are not important; my flywheel spins by inertia and without motor. For to tilt the spinning flywheel I pull the gimbal with a string instead to use motors.
When I pull the string for to tilt the spinning wheel, I cause a temporary precession. The same happens in the "nice precession demo".
When I stop pulling the string, I trigger a bit of temporary pure nutation, (the motion shown in my video above). The same happens in the "nice precession demo".
When I stop pulling the string,
the precession stops instantly.
In the "nice precession demo" instead the precession continues, which to me seems possible only because of the air drag of the fins.
Quite sure that the green flywheel was chosen in spite of the fins, as it has the largest AMI and AMI per unit mass of any one-piece LEGO wheel.
I didn't know this, and I trust you. This doesn't change the things anyway, the fins are there, and certainly produce some air drag.
Sorry, I think we're once again talking apples and oranges by mixing kinematic and dynamic viewpoints without due precautions.
I thought to this. Maybe who made the video used the term "precession" as a kinematic term, knowing that the motion is produced by the air drag.
But I suspect that, more probably, he simply used the term precession thinking to the motion of gyroscopes, for similarity of motions, maybe without realizing that the nature of the motions is different.