By spindle I mean the static axle which goes through bearings and discs. Designed supported at both ends or not it has no problem holding the weight at any angle. It's not important in the question.
I am confused by what you wrote here. What do you mean by static axle? Are you referring to when it does not spin? Either way, the fact that the spindle is only supported at one end is in fact very important. If it were supported at both ends, gyroscopic precession would not occur in a vertical drive. However, since it is only supported at one end, gyroscopic precession will take place in a vertical drive. I cannot see how that is not important to our discussion.
If the disc is not spinning it's axis can not be tilted out of spec by gravity itself, because gravity can not overpower the hold of the disc.
However, the gyroscopic inertia possessed by the disc spinning is stronger, and that force can be altered by gravity.
I am not sure how you arrived at the conclusion that gravity is too weak to tilt the disc out of spec. However, assuming that is true, why should gyroscopic precession tilt it? There are no extra forces acting upon the axis; gyroscopic precession is merely an effect, not a force. The force causing it in this case is gravity, and will act upon the spinning axis with the same magnitude as it would on a stationary axis. A rotating object cannot begin to exhibit gyroscopic precession merely by spinning; it must be acted on by some outside force.
Also, you incorrectly refer to gyroscopic inertia as a force. It is not a force, and it will effectively resist tilting of the axis, not encourage it. Gyroscopic inertia is the tendency of a spinning body to resist any attempt to change the direction of its axis of rotation; thus once the disc is spinning, it is more difficult to tilt the axis. If you say that gravity will not tilt the stationary disc, it is even less likely to do so when the disc is spinning.
When the axis of a spinning body is altered by an external force it's called gyroscopic precession.
Precession occurs because the applied force is not perpendicular to the axis of the gyroscope.
Your first statement here is correct, the second I cannot make a judgment about since it depends not only on the direction of the force but where it is applied to the system. However, I fail to see how either of these statements lead to the conclusion that gyroscopic precession would not occur in a vertically mounted drive. Gyroscopic precession is directly related to the torque on an axis non-tangential to its rotation. Torque is a cross product between the force on an object and the distance from the point of rotation. Thus when the angle between the force (in our case, gravity), and the distance vector (horizontal, in a vertical drive) is 90 degrees, torque will have its maximum value. Then gyroscopic precession will, in turn, be greatest.