With all the electrical problems you have been having sounds like a cascade failure, and the common point to all the electrical is the rectifier (which I mistakenly called a regulator in another thread). I am not sure how a jonway 2t is wired, but somethings work directly off the battery , some off the rectifier, so they only work when running.
Just as an example my blinkers and tail lights work when not running, but headlights only when running.
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