In my experience a Code 36 seems to be one of the most difficult to diagnose for the mere fact that when the Control Module sees something that it just doesn't understand or assign to a more specific code, it seems to throw a Code 36 at it. ???
Even on my personal Onan 10kW QD a couple of years ago I ran into the same scenario. The genset was shutting down just like I had turned the switch off, yet it kept throwing a Code 36. I too did all of the preliminary checks "by the book", but in the end I decided to use my head and go to the source and check the generator output just for shits and giggles. Sure enough, when testing right at the customer connection I had one leg putting out 120 VAC and the other leg 1.3 VAC. Bingo, now that I know it is not powerplant (engine) related I could see the scenario. The control module was not seeing the proper voltage on both legs so therefore removed the exciter voltage from the control module which therefore shut off the fuel solenoid.
Now when I work on a genset and don't have a more specific code to work with I go to the source and check the voltage output right at the generator and confirm or verify before going any further. Many times it is the powerplant causing the shut-down, however, just as many times it is not and a problem with the generator portion of the genset but the code would have sent me on a wild goose chase.
If voltage output is at or near rated output then start looking towards the powerplant, otherwise look at things like the exciter circuit, whereas on diesel gensets supply the power to keep the fuel solenoid engaged and on gasoline gensets supplies power to the ignition circuit. Other areas that may or may not be coded correctly are open or shorted stator windings or rotors (although rotors are quite rare). Brushes and slip ring issues can also throw an erroneous code on occasion.
Bottom line is verify voltage output then go from there.
Hope that helps.
Mike