Coleman Powermate 6875 Help (gx340)

I recently acquired a Coleman Powermate 6875 with a Honda GX340 engine. I was told the carburetor was bad and it was missing a few parts. I cleaned the carburetor and switched the main jet with a carb from another gx340 off a pressure washer for troubleshooting. I compared the floats and everything was within range. I purchased an air filter assembly and the manual choke which I noticed was missing. When I was installing them I noticed that the solenoid to the top right of the pull start does not have wires. The carburetor did not have a solenoid at the base of it either. I have searched a handful of hours for a model specific shop manual and an auto throttle assembly (I think?) and I have had no luck. I broadened the search and still nothing. Looked long into an automatic throttle on other models and I'm lost with wiring. What am I missing here?

I can get the generator started using starter fluid but it shuts down after. I removed the fuel filter for troubleshooting and that will get replaced when I have her running. I noticed the muffler is missing bolts to the cylinder head. I have attached pictures of what's all here, as well as where the butterfly is positioned with the choke on and off. I appreciate any and all help.

Thanks in advance,

Kyle