I presume you checked the output from the transfer switch. There should be 2 input lines, one from shore power and one from generator. The output of that switch is goes to the breaker box. Since you get power from the generator TO the switch. And since the output FROM the switch is common to both. The only thing that I believe can be the issue is the switch itself. If the internal relay contacts are damaged, even though the switch clicks, the output isn't being switched.I could certainly be wrong, but I had a similar issue in an earlier RV and had to replace the switch itself. Don
https://www.rvwithtito.com/download/Onan4000MaintenanceManual.pdfAdjusting the generator engine rpm changes the output frequency and voltage. See page 80, section 8-13, spec D and later. Try connecting your voltmeter to an outside 120v socket and measuring the voltage while tweaking the altitude adjustment knob (mixture screw) before messing with the throttle stop screw (governor).