[Big hammer]

A weak or poor connection can show voltage and not a good enough connection for power! At one point he had voltage on the movable point, and voltage on the fixed point, I think?
I agree that the problem is probably with the points, but a bad weak loose connection could be the culprit anywhere from the battery to the points. I had a no start checked for voltage at battery, fuse, terminal posts, coil, and the movable side of the points. All was good ! cleaned the points that looked good and went for a drive!