Onan generator only runs with switch held in

Some of the older units had a wire wound resister in the start circuit. The idea was that when pushing the start button it gave full voltage to the circuit to make starting easier. When the button is released the resister would cut the voltage back somewhat to reduce wear. If the resister burns out the circuit goes open when the button is released. If you have the resister you can check it with an ohm meter. If it had electronic ignition, you need to look elsewhere.