ShyTorque,
The difference is that helicopter rotor blades have flapping hinges, which alleviate the feedback into the rotational axis (the crankshaft of your aeroplane engine, or the main rotor shaft of a rotary winged aircraft) by allowing flapping to equality. Cierva discovered the need for these flapping hinges during flight trials, after he scaled up his small model aircraft.
I agree that the flapping makes things much easier on the mechanical components, letting the air take more of the forces that hammer from a wrenched propellor to the crankshaft bearings of a piston engine.
However, if you make a toy gyroscope with the ability to flap when torqued, it would behave just as the regular toy you're familiar with. I suggest that it's the exploitation of the power of the airflow around a rotor that makes the difference, not some sort of cancellation of the couples and angular momenta involved because there's a hinge.
Do you feel "gyroscopic" forces when a hand blender is in soup? Not like you do in air - because the coupling of the rotating blades to the soup is very strong.