The problem with the rigid rotor is dissymmetry of lift. Juan de la Cierva figured this out.
https://en.m.wikipedia.org/wiki/Juan_de_la_Cierva
So can you crank in enough cyclic pitch to overcome this dissymmetry? Let's say you could. For a US rotation helicopter you need a bunch of right cyclic as you start going forward. But if you allow a bunch of left cyclic you give a tool to the pilot to get in a lot of trouble. And then what about sideward and rearward flight? I guess the pilot needs some of that addional travel to account for it. But how do you limit his authority in the down flap direction when he changes mode of flight?
Flapping does all this for us and it works as a bias summed with the collective control. The stick does move laterally with airspeed but not really all that much that we can't live with it.