The two are really two separate systems..
You can plug in a lamp into one of the outlets and get the engine to fire up for a second or two.
The bulb should light up and from your description, it has nothing to do with generating electricity but the engine itself.
The spark plug should not be sooty and points to an overly rich ENGINE condition...
Before you try to start it, pull the spark plug, attach to the coil wire and ground to the head.
Crank the engine over and look for spark. If no/little spark, try another plug. You'll likely have spark.
Spin over with the plug out should the engine be flooded and check for excess fuel coming out of the spark plug hole, there shouldn't be any...
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