After fiddling for four whole days with this moped I finally got it running again.
I never did get the flywheel off, but I inspected every single wire in the wiring harness and I found some suspect connections which I repaired.
I'm a little confused as to why the wiring harness problems mattered since I had disconnected the engine from the harness completely and it still ran (or nearly ran).
I also have no idea why it was fouling plugs severely and then just started burning clean again.
Oh well, the moped now runs correctly.
Thanks Duck, Dan, and Ron for your input.