Whenever you're dealing with a somebody else's past wiring experiments, you basically have to start at step 1 and go over every component in the electrical system to see what exactly you're dealing with. My guess is P.O. swapped in a 12v lighting coil for the headlight and the rest of the system is still 6v, but you'll have to test to find out.
We're going over wiring at our free Moped Workshop LA class in Boyle Heights this Sunday the 25th. I know it's not exactly close to you but if you can make you can learn about moped wiring and if you're able to bring your bike we might be able to help you diagnose your issues if you pay the shop that lets us host the class the $20 day rate.
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