Most of these old twins won't run on a bad or low battery, much less start by a kicker or trying to bump-n-go them. I would assume they should, but because it does run, perspective from a seasoned Honda Head is needed.
Personally I am thinking the main key switch, I dunno if a regulator or rectifier could cause this for sure but it feels right.......the rectifier acts as a one way flow valve for the electrical system, but is technically speaking under constant voltage. When they go bad, they either let no voltage pass through at all, or the voltage is not stopped at all and thus a battery goes dead for "No Apparent Reason" in a day or two.
The rectifier only controls the Voltage Outbound of the charging system that is needed to run the lights without exploding bulbs, or charge the battery without boiling it. So in theory a disconnected or blown rectifier "COULD" possibly cause this issue as easily as a moody key switch.