Gets hot and stalls out

Now that you are home and the car will not start you need to start diagnosing what's not there

Do you have spark?
Do you have fuel in the carb?
If you have spark what is the timing. (With a helper you can do this while cranking)

Intermitant issues are a pain in the butt. The best you can hope for is it to happen where you can do testing.

I am more convinced it is not a direct engine heat issue

It could be an electrical heat issue like the ECU is getting hot just from usage.

Another possibility, while you replaced the alt there is a loose wire or skinned wire that under just the right conditions it shorts out.

The pickup or the wires to the ECU