I agree that it's primarily a charging problem, but a better battery is a big help.
Although titled as a 1969, the motorcycle was likely made in 1965. It has a flywheel mounted generator, and a rectifier, providing 6v DC current. Because it's flywheel mounted, the generator turns at engine speed, so at idle the system is consuming more power than it's creating. The battery size is limited by it's position in the frame, and is on the small side. The whole system is marginal, but this was common w motorcycles of that era, although they were soon replaced w better systems when electric starting became the norm.
Improving the charging would entail a complete gutting and replacement w a modern system. Part of the charm of an antique bike is running as much original equipment as possible, so I'd rather try to keep as close to original design as possible. Attached is the wire diagram, and original pages on generator repair.
I'm open to ideas.
Joe
P.S. The blue bike in my signature photo is a 1960's Sears/Gilera, which also had electrical issues steming from the orig design. This board (including you, Crustschow) did outstanding work in designing a 6v AC voltage regulator that turned the bike around in it's performance. It really changed it completely.