Since we are discussing the realm of the weird, let me add this tidbit.
I recently had a new battery in which the positive terminal was broken internally in the battery.
It would cause symtoms very similar to what is happening here. I stumbled upon this by accident, just going through the basics for the umpteen time.
I changed the battery and all was good with the world again...but it has symptoms of a circuit breaker resetting itself after it cools down.