Can you snap (or aileron) roll a helicopter?
In a teetering head machine, of course you can - once. In a BK, do it as a barrel roll, piece of cake.
The gyroscope effect is stuff-all. The way the blade reacts is simply Newtonian. Apply a force to the blade, you get acceleration, which takes time. While the blade is starting to climb (or fall) due to the aerodynamic force, it is also still turning, and by the time it reaches its highest or lowest point, it has turned approximately 90 degrees. The force in the original direction has reversed, and the blade starts to accelerate in the other direction, and away we go again.
There is no force being applied 90 degrees off the axis to create precession, the force is being applied through a full 360 degrees of travel of the blade in a sinusoidally variable amount, via the swashplate.
The force is instant, the resulting movement takes time. A high-inertia blade will react slower than a piddly little R22 blade - hence the 72 degrees of advance in that system - look up Lu Zuckerman's "missing 18 degrees" thread.
Try to spin around on the spot with a bucket in your hand, and raise your arm to shoulder level while spinning. See how far around the circle you get before the bucket is up there. Then put water in the bucket and see if the gyroscope theory works now.