Then there is something special about a rotor system that makes it different from other objects in the rest of classical mechanics
Yes, I think that is almost correct.... The rotor system is individual blades flying individual paths, the gyro is a solid, single piece.
The rotor system is not a special system in terms of classical mechanics, it is a different system because it is not a gyroscope. Thus the laws of gyroscope mechanics do not apply to the rotor system because the rotor system does not qualify as a gyroscope - at least the one we are talking about here.
Angular, or linear, bike wheel, or elephant momentum is still real, and it is still part of what is happening in the rotor system and the gyro....but it remains irrelevant to our discussion on precession.