I agree with the condenser. You probably have one laying around in your shop.
It's located behind the magneto when you pull the fan cover and fan off. Mag needs a puller to remove.
Points are hiding in there, too. Clean and check the gap.
You can hook up an external condenser and leave the old one in place. Hook up the condenser wire to the red wire (+) on the coil, and ground the body of it to the frame.
V1's have a brake light cut-out as a "safety" feature, where if the bulb blows out, the bike won't run, that's why it was mentioned in an earlier post.
It's the blue wire coming from the engine. You can unhook it from the terminal block and ground it to the frame to bypass it if to see if it's a wiring problem.
V1's are one of the simplest and most reliable engines, with tons of stock and performance parts available...