Don't think my Onan has a prime function either...at least when I press the button towards stop as recommended, nothing happens...can't hear fuel pump, and when you disconnect the fuel line, no fuel comes out until you try to actually crank it over. It usually takes me about 3 tries to crank mine too, and I usually have to hold the auto-choke until it cranks. Once she fires, it runs like a purring sewing machine.
I suspect the gas in the bowl evaporates between uses instead of it running back into the tank from the fuel line. The pump should have a check valve already on it. It runs based on demand from the carb which has a float switch that opens when the level in the bowl goes down due to usage or evaporation.