I wonder if you could:
1. Measure and monitor voltage to the solenoid as you press and release the start button.
2. Use a battery and alligator clips to give the solenoid 12v directly.
I always recommend using an empty water bottle and short piece of fuel line to gravity feed fresh gas directly to the carb. This eliminates fuel pump, fuel filters, fuel line, old gasoline.
Also allows you to feed a very strong mixture of carb cleaner into the carb if you suspect old gas has gummed up the jets. I have fixed clogged up Onan carbs many times using that method. It's not ideal. But it's quick and easy and cheap.
