I believe I can offer a theory about how the lift arm can get moved sideways.
It's caused by 3pt loads, despite the fact that heim ball joints only transmit force, not twist.
Looking at mcmxi's first picture in post #69, it dawned on me that the lifting rod and lower link pin are in a moving plane that is not perpendicular to the axis of the rockshaft.
Resolving the lifting rod force into components parallel to and perpendicular to the rockshaft axis, there is indeed a horizontal component. Its amount is the lifting rod force times the tangent of the angle by which the rod deviates from vertical (see markup). The stabilizers keep the lower arms from moving, and are not involved. The horizontal reaction is taken at the rockshaft.
Often the horizontal force gets masked by sliding friction within the fits, but lifting and lowering changes the force during operations. Its these moments the arm shifts a little on the shaft in its non-zero clearance, eventually working its way along, a little at a time.
A way to visualize this is with a trolley-type crane. If you push the load along, the trolley follows.