I suppose depending on how hot, vaporisation of fuel more likely than dirt in the idle jet as that would be constant. I have had ignition coils and condensor's that didn't like getting hot after a run, but generally a noticable misfire and probably wouldn't want to restart if it died.
Sorry just thinking about what you mentioned re generator light, what is the output when engine running normally, my mind was wandering off down paths of high loads taken to run a charging system with issues and then onto if you had electronic ignition was the voltage dropping down so low it affected the spark etc. etc. Another off the wall thought as an apprentice in the late 1960s a friend had a tuned Mini with a "Reece Fish" carb which gave extra performance but had the habit of icing up, so it would die and then on cranking it would restart as the ice would have quickly melted from the intake side of the carb, they brought out a heater kit for them in the end. Hopefully nothing that involved.