What fundamental mechanic causes a gyro to behave like a gyro, that’s missing from a helicopter rotor that behaves like a gyro?
The missing element is Rigidity in Space. Because of all the twisting, flapping, dragging and aerodynamic blade forces, the "disk" is dynamically unstable - and gyroscopes are not. It will not hold its position or orientation, it will diverge and crash, unless the coffee-stabilised pilot stops it from doing so, by playing with the cyclic.