The spinDup function per se, worked fine. But, if you can’t find the correct center of the circle, then spinDup only makes a mess of things.
After finishing the wheel with alt-S, (which the documentation erroneously says only works for curves), I went back and tried the spinDup method again, using your method of finding the center of the circle. It worked! I didn’t realize you could snap to a selection of multiple verts and it would find the center. So what I was doing, (select the circle, shift-D, esc, alt-M to center point, snap cursor to selection), for some reason was coming up with the wrong center point for my spinDup. Simply selecting the circle and using snap cursor worked much better for some reason.
Or, maybe I was selecting some stray ‘behind’ verts. That happens to me all the time, and it’s really aggravating. I wish blender would just select the front vert all the time instead of randomly selecting verts in back that I obviously don’t want. Here’s a screen shot of my wheel. You can see from the depth of the wheel how in front view it would be easy to select the wrong vert from a stack of verts and not notice.