This has been a real headache of a bike - but I am happy to say it now actually runs great now!
The moped really had my beat, most of the common troubleshooting (Cleaning the stock carb, cleaning points, installing new gaskets, etc) ended up not being what was holding this bike back. Given the lack of carbon on the piston, I am willing to bet this is what was the problem for this bike all along.
I've removed the decomp cable from the bike, since it comes in contact with the piston when you move it and would probably damage it again. Maybe that is what happened all those years ago.