There are two very common issues, one slightly common and some rare birds
Most common: On the generator there are usually 2 circuit breakers These are two position breakers, not the more comon (on/tripped/off) just on/off(tripped).
The ON/OFF markings are often a tad hard to see behind on the "Base" of the toggle.. (A flashlight may help shine the light on) and they may be in the OFF position.
Second: Has been covered above ("Generator outlet, manual transfer system, you plug into it)
Third (not so common: Faulty ATS.. (Automatic transfer system) Failed on mine, NOTE.. Many people will start the Generator while plugged in or, after a power fail, start the genny while plugged in and when power comes back shut it off.
Propercedure is to turn off big loads (Water heater, Space heaters, A/C, and the like, A/C's should be off for a minimum of 3 minutes, During that 3 minutes you change power sources.
Finally we have wireing errors, broken generators and other things.
ONAN generators (DO not know about other makes) are smarter than the average user,, If they are not making with the 120vac,, They think "Why waste fuel" and shut down.. So the fact it's running strongly
suggests the generator IS working.
Finally... Many RVers have a few screws loose... the kind that hold wires (loose nuts too, same kind).
Finally the Lights and water pump.. these are 12 volt devices.. Sounds like your house battery is not connected (STORE mode or AUX OFF mode).
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times