Agree with "probably fuel pump" but ours was not. Prev owners had let it sit without use/exercise. It would start, pick up the rooftop A/C as a load, and run as long as the load didn't drop off. If I set the A/C to Beyond Cold, genset would run till shut off. If A/C cycled out, to Fan Only, genset would quit. Problem was carburetor. The "idle circuit" was plugged with gum. Remember, on these RV gensets, Idle RPM and Full Load RPM are the same. 3600 in this case, but the Throttle Settings are entirely different. At Load, throttle is open and gas pulled through the large Main Jet. At "idle" throttle is closed and a tiny amount of gas has to get through small Idle Passages.
But a Fuel Pump is subject to failure and an Amazon replacement at $50 is 1/4 to 1/3 the price of an ONAN carb. So if I was throwing parts at it, Pump would be my first toss.
You've heard from OP's who have had ONAN fuel problems that went the two common ways. Be reminded the Model and Spec Code are very important with an ONAN. Always post the full Model Number. The Spec is the letter at the end, "K" in CD's case above. Model/Spec will get you into the subtleties of a long run of these gensets. For example, people will shout "low oil switch" and duck behind a rock. Fact is, ONAN deleted that feature from 4KY series before production of the "K" spec units. That means that switch is a bunny trail in your case.