If the symptom is that the starter does not operate with the key, look to the starter solenoid. Starter heat soak generally takes much more than just a few minutes of operation. Also check to be sure the starter is bolted down tightly and that battery to frame and frame to engine and frame to body grounds are good.

None of the fuel, spark, alternator or other issues will cause the starter to not crank the engine over.

I know a Dan Weaver in Oakland County...any chance it's you?