Look at the L bracket that the rubber wheel slides left and right on.
Get a long flat screwdriver and wedge it between the bracket and the tube that the shaft slides left and right in.
Now fire it up and have the wifey shift it to low speed and u bend the bracket back a little and see if the wheel meanders back to high speed. Bend it more and see.
If the wheel went from high speed to low, then bend the bracket toward the back of the mower.
What I have seen, as the bracket is bent, the shaft is not directly over the pivot point of the drive plate so the energy is not perpendicular to the wheel but is instead dragging the wheel from High to Low speed(outer to center) or Low to High speed(center to outer) depending what direction the bracket is bent to.
Also, u can check the bearing of the drive plate, if it is shot and wobbling alot that could cause the same problem. It looks well kept so i would bet it is just bend the bracket to fix the prob.
The spring keeps the pully apply pressure to the disk but it shouldn't pull the assembly to the middle as you see with the millions of mowers made with this design.
The bearings are okay from what I can see. No undue jumping around ect...
The driven rubber disk I think is centered on the drive disk.
That picture I found on the net. His mower, in that pic is set on max walking speed or all the way to the left.
Again when I slow mine down, rubber disk moves to the right. That big silver spring tugs it back to a faster speed. No detents in the speed lever or cable. Has the red knob on the speed lever, fits in the palm of your hand.
Wondering if someone replaced either pulley and now the belt turns the drive VERY fast? That drive belt will turn so fast you can't read the writing on the belt. As in drill press speeds. As in much faster than I've ever seen on a pusher.
That L bracket you speak of. That has the speed selector rod attached to the L part right? And you want me to tweak that tab left and right?