Is it possible for the slow starting bikes that have the original rectifiers on them that the goat is heating up the rectifier that while it is running it will run but if the bike turns off that the rectifier is too hot to produce what is needed to start the bike?
Maybe, if the heat is causing a drop in charge voltage and the battery is down before the engine is shut off.
But the heat soak after shutdown won't matter if the battery is charged, the r/r won't be in play until after the engine starts. Heat soak could affect a weak battery, though not as much as cold temps would.
Re: your post about r/r removal ... Should be a post or two on it, my old one is still in place, new one under the left side cover. Screws looked pretty corroded. Break the heads off the screws with vise-grips? Drill the heads off the screws? Will it come out with the battery box? I have no idea.