Kryton, in that demonstration you're showing there, the angle the gyroscope is leaning at does not change. Thank you for proving my point. You will note that it is only turning along 2 of its 3 local axes. The third axis, which is offset from the direction of gravity, is being prevented from rotating. If you look at the animated gif and the wheel, you will note that its angle in relation to gravity is not changing at all. So your gif, which is supposed to prove me wrong, only serves to demonstrate the very point I was making.
Gyroscopic precession does NOT cause the front wheel to turn toward the bike's linear angle of motion. IF it did, you're talking about it never being able to right itself.
Gyroscopic stabilization relies on Newton's Second Law of Motion, specifically in regard to preservation of angular momentum. The faster the wheel spins, the more force it takes to change the angular momentum. Once again, this is why bikes are harder to lean at a high velocity relative to the ground than at a lower velocity relative to the ground.
Bikes leaning toward the vector of motion happens at any speed. It happens at speeds at which any gyroscopic forces applied to the system are too weak to move that much inertia. Remember, it's about momentum, not about simple s rotation. Before any gyroscopic force, precession or otherwise, can even take effect, it has to have enough kinetic energy stored in the angular momentum of the wheel that it's capable of moving the mass of the bike. Again, bikes are harder to lean at high speeds but easy at low speeds.
It is extremely easy to demonstrate that gyroscopic precession does NOT cause the wheel to turn when the bike leans. All you have to do is pick up the bike with nothing spinning and lean it. This is a lot easier to demonstrate with a bicycle, obviously. Pick up a bicycle hold it level with the ground. Now lean it. The front wheel will turn toward the lean angle.
Did you do it? If you did, you just demonstrated the wheel having that behavior occurs even WITHOUT gryoscopic precession, and you demonstrated it by having the wheel not spinning while you did it.
As you know from having enough experience to have a casual understanding of counter steering, that if the wheel tries to do that in a lean, while moving, the bike is going to right itself... and that behavior is present even when the wheels aren't turning.
Here's a video demonstration:
If the behavior occurs even when the wheel is not spinning, gyroscopic precession is completely ruled out as the cause of the behavior.