first, on the GL1000 there is a ballast resistor that reduces voltage when the bike is running. It is bypassed when the starter button is pushed. If the resistor is bad or unplugged the bike will do exactly what your quad is doing.
second, it could be the starter button. I don't know exactly how the wiring goes on that model but it could be something like the Honda motorcycles that disconnect the headlight when the button is pushed.