Sun of Atom, it seems your Instructor Training was lacking a lot in aerodynamic theory - not your fault, it is the result of too many junior instructors passing on what their junior instructor told them. Chinese Whispers.
My understanding was that it was gyroscopic precession when an outside force is acting on the disc, and phase lag when we are changing the disc itself through control inputs. You're saying it's always phase lag?
GP doesn't exist in a rotor system. A gyroscope is a flat disc fixed rigidly to a shaft. A rotor system had flexible blades attached to the shft by flapping hinges, and often lead-lag hinges, and the blades can feather as well. Too many flexy floppy bits to be called a gyroscope. But to make it easier for students to understand phase lag and lead angle, it has been likened to a gyroscope. Sadly, some sources have locked onto this part and pass it on as the truth.
If I'm understanding this correctly, flapback is moment of dissymmetry of lift, and the rotor system needs a nudge from us in order to move back into symmetry. Once in forward flight, it's actually a control input that creates a moment of lift dissymmetry, after which the rotor system will re-balance?
The only time the disc is OUT of symmetry is the momentary change of something, be it a wind gust or a control input. Three poofteenths of a second later, the lift over the entire surface of the disc has equalised, otherwise you will get acceleration in the change of disc attitude.
Say you are in a stable hover, cyclic neutral. A puff of wind blows onto the nose. The initial part of flapback is the disc flapping away from that extra wind. The disc has "flapped to equality" and lift is again equal over the disc, but it is now pointed backwards, and the aircraft starts to move backwards. the momentum builds, the disc passes from having a breeze from the front, to now having the effective breeze coming from the back, and it flaps away from that "breeze". The fuselage lags a bit behind, so as the disc moves forward, the fuselage swings nose down, helping the disc to go a bit lower at the front. The forward motion speeds up, and again it flaps away from the wind, to go nose high. The lagging fuselage swings even further nose high, so now you have an exaggerated rearwards thrust vector. Very soon, you will crash.
To stop this, as soon as the pilot sees the attitude change from what he/she/they/whatever desire, a cyclic input is made to stop it. And in
stable flight, be it a hover or forward/sideward flight, THERE IS NO DISSYMMETRY OF LIFT.
So the large forward cyclic input during takeoff is, at least in part, due to the weight shift of the fuselage as it swings under the moving rotor disc, plus the changing angle between the fuselage and rotor disc altering our cyclic inputs?
To move into forward flight, a cyclic input is made to lower the disc (and the nose) at the front. The aircraft starts to move, and with movement comes the flapback (countered by progressive forward cyclic) and inflow roll (countered by lateral cyclic). For every increase in speed, there is an initial forward cyclic to start it, and then some more to hold the attitude down against flapback. Eventually you reach the limit of cyclic movement and you cannot go any faster because you cannot prevent the subsequent flapback. The weight shift of the fuselage, as you put it, isn't a factor in this, unless there is some extreme drag associated with the fuselage.