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.