+1 on the ignition switch theory. My one skill here is electrical. A solenoid is just a set of heavy contacts that close to increase the amps to the starter motor. To operate at all, the solenoid needs the 12V from the ignition switch. If that switch is open, the solenoid has no voltage.

If your ignition switch stayed closed (despite the key position), you would continue to get the cranking.

I don't have real insights on the original "hot starting" behavior, but I would rule out spark by squirting a little starting fluid down your carb the next time you get into a "hot start" situation and before cranking it. If you've got good spark, it should try to fire up immediately. In that case, I'd start suspecting fuel issues. If it just cranks on the starting fluid, I'd go down the ignition/spark track.

Best of luck.
Tom