There's a reason the makers recommend these small generators be exercised once a month. The tiny jets in the carb can get virtually to completely blocked in a few months of non-use. Just a spec of varnish from evaporated fuel will make it run rough. On larger carbs the jets are also larger and you can get gumout or other carb cleaner through them to eventually clear them. Difficult with the tiny jets - just can't get enough cleaner through them to melt the varnish.
Good luck in finding a rebuild kit. Don't think they are available - even after market. If the carb clogs they want to sell you a complete replacement carb.
There are previous threads on this subject. Some have installed fuel cut-off switches to allow them to run the carb dry (can be hard on the fuel pump); others have installed a "y" or "t" to flood the carb with cleaner before shutdown.
Can't explain the choke linkage issue.