Vessbot - I get the impression you know a lot about physics but very little about helicopters.
Phase lag is governed by Lock number which is a ratio between blade inertia and aerodynamic damping.
A swash plate does not impart an instant or impulse force to a rotor, it builds gradually as a result of turning a rotating motion into a vertical one because you have to change the pitch of the blade to make it climb or descend.
No gyros I have seen have hinges that limit how much the movement of the mass of the gyro can be transmitted to the hub - a rotor system does.
Just because systems exhibit similar behaviours doesn't mean they have the same cause.
The lock number has very little influence on the phase lag. (It is also not the delta 3 angle, as somebody else claimed)
The phase lag is predominantly determined by the flapping hinge offset. This is the ratio between the distance from the rotor axis to the flapping hinge and the distance from the rotor axis to the blade tip.
Then you get two effects that occur simultaneously and therefore overlap:
1. The gyroscopic effect leads to a maximum deflection at 90° phase shift from the maximum lift force. This effect does not affect the orientation of the rotor shaft. This means that the rotor blades rotate about a (slightly) different axis as the rotor shaft.
2. The non-zero flapping hinge offset leads to bending moments in the rotor shaft due to the deflection. This leads to a rolling or pitching moment of the entire helicopter, which then turns according to the magnitude of this bending moment and its inertia. This leads to a change of the orientation of the rotor shaft axis. The phase shift of this effect smaller than 90° depending on the hinge offset.
These two tilts of rotation axes combined lead to a combined phase shift, that is smaller than 90°.
Note that also flexbeam rotors have a hinge offset, which is determined for the equivalent ratio between blade deflection and bending moment at the rotor shaft.