Spark Enigma

It is a pretty simple circuit. If not the coil or ECU, it is likely in the key switch. The way the switch behaves could depend on current heating. By leave in "on" position, do you mean the "run" position (oil light on) rather than "accessory" position. You don't want to leave the ignition powered long with the engine not running since I read that can destroy an ECU or points.