Where did the OP say it was a Honda?
Does Honda sell a 2800watt inverter model?
The posted info about Honda carburetor simplicity is all good,
And I think most Chinese built generators have similar if not identical carburetor
I just don't see anyplace where the OP said it was a Honda
I know my champion inverter generators have stepper motor controlled carburetor, not throttle linkage governor controlled carbs
At very low loads and econ 'on' the 2000i sounds like it is hunting,
The problem is the speed control programming
The first step up in speed is more than what is needed, but it's not happy running 'in first gear idle mode', so it goes up then back down repeat to infinity
And not a dang thing to be done about it
It's let it be, or play with the choke,
Or throw on more load, like a small fan