The troubleshooter says control board. I could rip apart the carb, take apart the stator which is probably well rusted after seeing the flywheel, but I don't have any more time or money to flush on it.
I think it great that you came back to tell us how things turned out.
I ran into a similar situation a few years back and have no idea how things turned out. The people had purchased a new generator and then let it set there for about 6 years. I tried to get the generator shop that sold them the unit to come and do a start up on the unit but they would only do so if the unit was hooked up legally and correct and the people didn't want to pay for that so that was the last I herd about it. This was for a commercial business and they had the generator installed by a handyman that didn't have any idea what he was doing and as far as I know they have never gotten any service out of an expensive generator because the were to cheap to have it installed properly.
I for one have no idea how long you can let a generator just sit there and rust before it's not feasable to repair. Sounds like you gave it a nice try. There is a time when you get tired of throwing good money after bad and just break down and get a new uniit. I think it's probably a wise decision.