If your seeing less than 12-14v to the battery when running depending on engine speed then you have either a bad regulator/rectifier or a bad alternator/stator coil. Most likely its the regulator/rectifier as i have replaced them fairly often.
Take a voltmeter and set it to AC Volts and check the two Yellow wires coming out down by the recoil, they are up to the frame tube by the sub trans shifter. Put your meter leads on the Yellow wires, doesn't matter which gets the positive and which gets the negative since its AC at that point. With it running you should see around 10-30v depending on engine speed. If you do then the Alternator/Stator coil is fine and its just a bad Regulator/Rectifier causing the issue.
Also, you should be able to start it with the recoil regardless of the battery and charging system. These have two separate systems, ignition and battery charging and lighting. They are completely separate systems electrically and have no impact on one another.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....