I How to quantify gyroscopic precession torque?

[A.T.]

...an internal roll torque reaction to yaw "cancels" an external roll torque...

"Internal torque canceling an external torque" is exactly the kind of confused talk that leads nowhere.

This doesn't change the fact that there is only one external torque that results in precession.

External torques and forces are all we care about.

The gyro initially "drops a bit", and only after the drop has started does the gyro begin accelerating in the direction of precession.

Because of angular momentum and energy conservation the gyro cannot just start precessing, while maintaining exactly the same spin rate and spin axis inclination. The precession motion itself has its own small angular momentum component around the vertical axis, which must be reflected by an equal but opposite change in the vertical angular momentum from the spin.

And it is that small angular momentum component around the vertical axis that you need to remove to stop the yaw-precession.

 

[A.T.]

The torque from the straw is smaller than the torque from gravity, yet the gyro "drops" at a faster rate than it precesses due to gravity alone.

Here a linear momentum analogy that might help to understand why it drops so fast:

Linear momentum:
The force of gravity acting on a body in circular orbit continuously changes direction as the body moves along the orbit, so the linear momentum transferred by gravity doesn't accumulate, it just goes into continuously changing the direction of the bodies linear momentum. When you prevent it from moving tangentially the force of gravity has a fixed direction, so the linear momentum transferred by gravity quickly accumulates, and the body drops to the center fast.

Angular momentum:
The torque from gravity acting on a horizontally precessing gyro continuously changes direction as the gyro precesses, so the angular momentum transferred by gravity doesn't accumulate, it just goes into continuously changing the direction of the gyros angular momentum. When you prevent it from precessing horizontally the torque from gravity has a fixed direction, so the angular momentum transferred by gravity quickly accumulates, and the spin axis drops down fast.

 

[rcgldr]

angular momentum ... gravity

Doing the math from the Wikipedia article, if there was no gravity, and instead the axis support allowed yaw rotation, but not roll rotation, then if a yaw torque was applied to the gyro until the same rate of yaw as gravity induced precession, then the induced roll torque exerted onto the support would have the same magnitude but opposite direction from the torque that gravity would exert if present. Once that same rate of yaw was reached, the yaw torque would go to zero, and there would be a Newton third law like pair of roll torques: the gyro exerting an "upwards' roll torque onto the support that is preventing roll rotation, coexistent with that support exerting a "downwards" roll torque onto the gyro.

The support would need a reactive external torque to prevent the roll rotation, such as being attached to some massive object. Once the yaw torque was removed, then the gyro's change in angular momentum would be balanced by an opposing change in angular momentum of the massive object, and the angular momentum of gyro and object would be conserved. In this closed system, there are no external torques, since the total angular momentum is not changing. This what I meant by the "internal torque" that opposes gravity in the original case.

 

[A.T.]

"downwards" roll torque onto the gyro.

Calling the roll torque "downwards" is a bad idea, even if you put it in quotes. That torque vector is horizontal, and continuously changes direction in the horizontal plane during yaw-precression. This continuous change of torque direction is key to why angular momentum doesn't accumulate during yaw-precression, but does accumulate when yaw-precression is prevented (see post #32). Calling that torque "downwards" doesn't allow to distinguish these two cases, which makes it a particularly bad naming choice here.

... This what I meant by the "internal torque" that opposes gravity in the original case.

Whatever complicated rationale, based on comparing different scenarios, you had for this terminology, it just sounds dead wrong, to say that an internal torque balances an external one for a specific scenario. Just like with your "downward torque" above, your terminology choices make it really hard to follow you.

 

[rcgldr]

Calling the roll torque "downwards" is a bad idea

OK, but the key point was the torques involved with a gyro precessing at some specific rate about a vertical axis is the same for the gravity case or the support that only allows rotation about a vertical axis case, and in the support case, the yaw rate which is the same as the precession rate in the gravity case coexists with a roll torque that opposes the roll torque from the support, and the roll torque from the support is the same as the roll torque from gravity in the gravity case.