For a beast of a wheel this sure does seem strange. The head or shaft is warped and needs to be resurfaced. I did the pen test where you take a pen / marker and slowly spin the wheel and touch the pen to the head and just barely keep it there so it shows you where the dips and raises are. There are multiple spots that are warped.
So I stripped an allen screw out trying to get the head off and so I had to take the whole wheel, shaft and kick plate off which is probably the right thing to do if the shaft is warped anyways. Off we go to CRM Fabrication and Repair. Heck of a shop with outstanding service!
Speaking of CRM; I am working with the owner on a prototype Chinese style hand driven portable wheel. A new type that can be used by my son who is too small for the full size kickwheel to sit on the ground with or to put on a tabletop. You will also be able to use a dowel in a hole on top to power the wheel. An example is shown here where the late master-potter Shoji Hamada is creating momentum on the wheel for another pull. So stay tuned!