I have sometimes felt that a thought experiment based on your point (2) would clear up much confusion and help to introduce the gyroscope in a better way. Rather than thinking of precession as an effect of gravity's torque on the spinning gyroscope, let's think of it as a given angular momentum (which of course doesn't need any torque to sustain it).
To start with, imagine that the horizontal axis rod is not pivoted, but rather is locked to a vertical shaft, which in turn is free to rotate around a vertical axis. But we have a torque sensor at the locked pivot, so we know the net torque that is trying to tip the gyro axis down. And let's start with the gyro not rotating either. Obviously, in this case the torque sensor will show only the gravity-induced torque, and will continue to do so whether we (manually) set up some "precession" type motion or not.
Next set the gyroscope spinning, with its axis rod stationary. Again, the torque sensor will still read just the gravity induced torque. Now, give the axis a little nudge to set it revolving around the vertical pillar. Once we give it that nudge, it will continue to revolve through sheer angular inertia. The word precession would be somewhat irrelevant at this moment, being a descriptive label rather than an explanatory term or a result of gyroscopic behavior.
At this point, the torque sensor would read less than before, because you now have your W-N effect. We can nudge the axis a couple of times more, and the torque sensor will read less and less as the gyrosopic upward torque increases with each nudge. At some point we can get the torque sensor to read exactly zero, and we now have the gyroscope "precessing" with some corresponding angular momentum. Again the "precessing" is just a descriptive term, not an explanation of why it is rotating. It is rotating because we have been nudging it, and now it has to continue to rotate because of inertia.
Finally, we remotely unlock the pivot while the thing is still "precessing", and we know it is going to remain horizontal because the torque sensor is reading zero torque. So in this picture the precession is the cause of the upward torque that compensates exactly for the gravitational torque. And the precession is the effect of the past history of the little nudges that we have been applying.
