Why do some helicopters have such complicated flight control mixers?
Flight dynamics of a helicopter are complicated indeed. Just to illustrate, consider a helicopter in a stable hover, with the pilot holding the collective, cyclic and pedals in the position that maintains attitude, altitude and heading.

They now want to gain altitude only, in a climbing hover. The sequence of events:


Pilot pulls on the collective, all rotor blades increase angle of attack and the rotor creates more lift. The helicopter climbs.
But because the AoA of the blades have changed, the engine must apply more torque to the rotor and the heading starts to change. So the pilot must apply pedal to change tail rotor thrust and compensate for the torque.
The increased thrust of the tail rotor causes lateral drift. So the pilot needs to input lateral cyclic in order to compensate.




To gain a pure vertical climb, the pilot needs to carry out simultaneous deflections of three flight controls! This collective-to-yaw and collective-to-lateral coupling not only occurs in the hover, but also in forward flight. When increasing speed, the pilot needs to:


deflect cyclic pitch forward;
increase collective for the additional required power;
deflect the pedal for the increase in rotor thrust - but at speed, the vertical fin and the tail rotor take over most or all of this;
deflect the cyclic to the side to compensate for the increased tail rotor thrust, or for the sideslip from the vertical tail stabilisation.


For small, simple and low cost helicopters, this is indeed the way they are flown. But for larger helicopters with a mission, some of the cross couplings are taken care of by the mechanical mixer: when increasing collective, a mechanical link automatically increases the inputs for the cross couplings. It does so in series with the flight controls, which do not move and retain their former selected position.



The drawing in the OP is of at the S-76, a medium sized helicopter, and we can find the following:


Collective/Roll mixer (lateral lead). Provides automatic rotor sideways deflection as a function of collective stick deflection.
Pitch bias actuator. Extends as a function of airspeed, so that the cyclic pitch stays near centre, for more comfort at high speed.
An unlabelled connection rod between collective and pedals.
Collective/Pitch mixer. This helicopter will start to pick up speed when collective is deflected, for a straight-up vertical speed, the pilot will need to deflect the stick backwards.


Also in the OP drawing: optional AFCS servos, which reduce control forces and provide the static stability that helicopter aerodynamics do not provide. Plus trim actuators, which change the neutral position of the feel springs or completely eliminates feel forces. And a yaw damper.

Very complicated indeed.