I have worked on a lot of old cars and found the points to be the problem only twice. Once when the points had actually broken and fallen off and the other when they were carboned up.

This is easy to see and fix. The contact part of the points will have a blackened or uneven look to them. A small file, known as a points file, is just inserted between the contact points and a bit of the roughness is filed off.

Points files were once (maybe fifty years ago) a common tool. I guess now the best way to find such a thing is to use a fingernail file.

And should the points be checked for dwell after filing them? Of course. But only if you want your engine to run as it should. If you are just wanting it to run it should start right up. But without adjusting the dwell it will never run smoothly.