If you think about it, this explanation cannot be true. The main force the helicopter exerts on its surroundings is along the axis of rotation. If you tilt the axis of rotation significantly in any direction, the direction of the force will change significantly and the motion of the helicopter will also change significantly. Again, this does not happen.
It's unclear what you mean by the "this" in the "this does not happen"
However ...
A helicopter changes direction by using the cyclic setting on its rotors. Then you get more lift from one part of the rotor cycle and less from another. That causes the plane of the rotors to tilt, and then the helicopter goes off in the appropriate direction.
For example, having more lift in the rear portion of the cycle and less in the front means the helicopter gets tipped forward, effectively like lifting one part of a plate. The down-draft now has a significant rearwards component, so the rotors get pushed forward, taking the helicopter with it.
If you then change the cyclic to "flat" so there is identical lift everywhere, the weight of the helicopter dangling from the rotors causes it come come back upright (after swinging a bit)
It's a weird experience sitting in a helicopter when it's doing this, especially if you're mostly accustomed to ordinary aeroplanes.
I don't see any contradiction in any of this, and it really doesn't seem to have any bearing on the fact that contra-rotating disks have no net gyroscopic effect.
When you say "this" you mean the translation of the pilot's controls into the angles of the blades at the various stages of their journey, mixing collective (giving overall lift) and cyclic (giving pitch and roll)
