Update:
I think the sprocket is actually OK, I think the torque on the bolts was distorting it slightly.
After I got the sprocket off, I cleared all the garbage out of the rim.
Then I found out the previous owner had rednecked the hub assembly.
There's probably a dozen stacked washers to space the sprocket.
Without it, the tire sidewalls make it impossible to mount a chain, since the hub needs to be longer. The mounting bolts are mismatched with how much space there is between the head and the threads, so I am hoping to get to the hardware store soon to get new bolts and maybe nuts. At this point, I'll probably just restack the washers to get the proper chain alignment.