Ours also uses house batteries to start the genset.  When we first bought our coach, we also took it for granted that the genset would run....it didn't. It would turn over, but would not run.  After some serious diagnosing, I came to the conclusion that the carb was varnished from old gas and the needle valve was stopped up.  Run some SeaFoam through thorugh the gas and see if that cleans the carb enough to get it running good.  If you have to, disassmble the bowl/needle valve and soak them in straight Seafoam for about an hour and then clean them with a stiff brush before replacing.  Onan will tell you that you should replace the carb, but you can take it apart and reassemble successfully.
Then run Seafoam in every tank to treat and clean the fuel system from all that crappy gasoline we are forced to buy now.  Unless you travel a lot, that's a bunch of gasoline just sitting in a tank going bad if not treated with something.