You are doing things mostly right but it certainly is frustrating.

There are a few more things which can cause trouble but I'm thinking it may have something to do with the distributor.

1. check for continuity of the small wire between the points and the coil. It should have less than an ohm resistance.
After checking it wiggle it a little and check again.

2. The advance mechanism in many distributors is controlled by fine springs which resist the pull of spinning weights. Things get strange if a spring breaks.

3. It is far less trouble to check the ignition timing than to take the distributor apart. You have to adjust the timing in any case as you worked with the points. Check the timing.
It is also under vacuum control and that hose should be squeezed shut when testing timing or the numbers will be way off.

Electrical problems are usually on or off,but sometimes they be intermittent.

The 30 second runs do sould like a fuel-related issue as they repeat so evenly. It can run a little while on a floatbowl of fuel, but not very long.

Valve timing chains can jump and make an engine stop, but they usually don't change back and forth so the engine runs and stops again and again.

An anti-dieseling solenoid is supposed to shut it off.
So if it was acting up it would be a prime suspect. If there is one there, remove it, remove its "pin" and replace the solenoid valve.

For the short time it runs, does it run normally?

Persist. You will solve it.