If it WAS a gyroscope, we would not need swash plate, cyclic feathering, stabiliser bars or such, just a small servo that pushes against the mast EXACTLY 90 degrees ahead of where we want the disc to point. But this doesn't happen.
I guess your passengers without the banged heads are paying for safe travel, and not for an accurate description of the nature of the machine they're in and how it works. However, perhaps they'd be more comfortable still if their pilot could give one.
That DOES happen, but it's not a small push. The last question above is the key.
If you did make a small push, you'd get an exceptionally small angular acceleration in response, rotating the plane of the disk in a direction 90 degrees away from the moment of your push.