Gen shuts off unless “Start” held down

your Onan 4KY spec H does not have a low oil level sensor, nor does the control board have fault code diagnostics. OP is probably correct, that generator is not producing AC power, which control board has to "see" to turn on run relay on board. Most common causes are bad field brushes or bad brush contact with rotor slip rings, bad voltage regulator, or bad control board. Worse case scenario is gen. rotor or stator shorted or open winding. Getting to field brushes is somewhat involved, as the fan/flywheel needs to be removed for access. Voltage regulator and control board are expensive items, so throwing replacement parts at it in an attempt to fix it can cost a bunch of $$$ and very possible would not work if brushes/rotor/stator are bad. Replacement control board and voltage regulator are available aftermarket from FLIGHT SYSTEMS in PA. Best bet is to find a qualified ONAN service to accurately diagnose problem. Good luck