Steve, please do try the "3 second" and "10 second" test, and let us know what you discover. It would be interesting to find that the amount of time in "Prime" mode actually tracks to the starting performance--because it really should not.
There are a couple of things I can offer based on experience. I've done quite a bit of small engine work in the past--replaced, cleaned, rebuilt a lot carburetors on a variety of engines. I've also had occasion to go into my 2019 Onan generator carb twice--when I've let it sit too long without exercise and the engine refused to start.
The Onan carburetor is a conventional design with the float in the bowl closing a needle into a seat to stop the flow when the fuel reaches the proper level in the bowl. As such, if there is no problem with the needle/seat, the carburetor will not "flood" regardless how long you hold the switch to activate the fuel pump. Pump pressure is not nearly high enough to overcome the pressure applied to the needle by the raised float. As a matter of fact, once the engine starts, the electric fuel pump is energized full time, and pumps against the needle constantly. Fuel delivery from the pump is much greater than the amount used by the engine when running, so essentially the pump is pumping against a mostly closed needle all the time--fuel flow into the bowl is only allowed to the extent that the float drop opens the needle.
There is not a "fuel return" circuit on this carburetor that flows fuel through the carb back to the tank like on many 2-cycle engines. The purpose using the prime switch is only to make sure that the carburetor bowl is completely full of gas and there is no air in the line or the carb. That being the case, the amount of time you hold the prime switch really doesn't matter, as long as it's long enough to completely fill the line and the carburetor bowl. I, too, have had instances when (after priming) I've had to crank the generator two or maybe three times to start when cold, especially in cooler weather. I think that is more a function of the choke operation on the carburetor than anything else. Conversely, multiple same day starts normally do not require priming for me--the generator usually busts right off if it's been run earlier in the day because the carburetor bowl is already full.
On the subject of Onan carburetors, I will add that I have found the "failure to start" symptom that occurs when you have let the generator sit for too long (3 months) without exercise track specifically to the main jet in the carburetor. Each time I've had to fix mine, I've found that the E-10 gasoline had deteriorated some in the bowl, and while the bowl and fuel shut-off solenoid in the bottom were OK, the main jet and adjustment needle had become "gunked up" from the deteriorated fuel, blocking any fuel flow through the main jet. The condition is not difficult to correct by just pulling off the carburetor bowl. You can pry off the altitude adjustment stop, remove the needle, clean the needle and the jet, and reassemble. Flush a little fresh gas through the line with the prime switch and it's as good as new. Takes about 30 minutes.
Mike
