Sounds like you, my friend, have a dizzy issue. My car did exactly the same thing.

Read this:

https://board.moparts.org/ubbthreads/ubbthreads.php/topics/1861900/re-no-spark.html#Post1861888

No, after it starts it runs great continuously. How do you know if the starter is the problem? I checked the ground at the ECU.
The last time it wouldn't start I added an extra ground. It still
wouldn't start.

The difference here is that yours would shut down after running
for a while. Mine runs continuously once and if it starts
whether it's hot or cold

If it won't CRANK - it's a starter issue. From your original description people could not tell if it would not crank, or if you had a no spark condition. It sounds like you have an intermittent no spark condition. With the key in the START position you bypass the ballast resistor and crank the starter - full 12 volts to the coil. When you release the key it returns to the RUN position and the juice to the coil is reduced by the ballast down to 9 volts. I would put the key in the RUN position )don't crank the motor) and check the voltage with a voltmeter first at the coil and then back at the ballast. The re-check the voltage while jiggling the wires going to the coil and then to the ballast - see if the voltage goes to zero - if so, loose connection somewhere or a break in the wire. Check the ground too on the ECU - the factory used a self tapping bolt with a star washer like the ones on the starter relay to cut into the steel body of the ECU. If all that is good check the gap at the reluctor in the distributor - .006 - .008. The DC distributor, if it's the older style, is really a factory distributor and will have the set screw and adjustment slot. If all that fails replace the pickup in the distributor - the thing you set the gap on.

I've come across this twice this year.My 70 Chrysler 300 with a old MP conversion kit...every so often it would not start but would kick when i released the key.Changed nearly everything...still did it.Finally took the P4120505 control box and swapped it with a crusty used blue box...been mint ever since.Customer towed in his big block Charger a couple of weeks ago with the same problem.Put a new Chinese control box on and no issues since.20 or 25 years ago the orange boxes had a similar issue...run fine till they warmed up and then would quit
until they cooled down.

I put a new ECU control box on and it still would not start. It
never quits. If it gets started it runs like a new car even all
warmed up.

It cranks great....just that sometimes (not always) it won't
start. Your idea is a new one. One that no one has suggested and
one I need to try the next time it won't start.

Next time it won't start, immediately disconnect the distributor connector and measure the resistance across the two contacts of the distributor plug.

If it's over 1000 ohms, you'll have likely found your problem.

What's a distributor connector? Can you describe the 2 contacts
of the distributor plug?

its the pigtail coming out of the side of the dist (has 2 wires and a zigzag pull apart connector). You would ohm the metal male tip and the recessed female terminal on the dist half of that connector

My problem was identical to yours. Too wide a gap at the reluctor. Thanks to your help I discovered the problem. Many....
many thanks for your input and thanks to everybody else that made
suggestions. All you guys are super!

Jim

Glad I could help, I know how frustrating it is!! Mike