The patient is a 1000W Yamaha EF1000is inverter generator. The 50cc four cycle engine drives an alternator that powers an inverter which outputs clean 120V 60Hz power.
In normal "economy" mode the speed of the engine is controlled so the alternator's ouput is adequate to maintain output power. With the economy mode off the engine runs at constant speed, about 5000 RPM.
The engine starts easily and idles smoothly. A little puff of black smoke when started, exhaust seems clean immediately after that.
By applying load slowly with a variac the generator ramps up and runs fine 'til about 50% load, after which it dies.
A relatively small load will kill the engine if it's applied quickly, regardless of whether economy mode is on or not. The manual recommends turning economy mode off for loads with starting surges, e.g., motors.
What's been done: Cleaned and gapped the plug which looked a little dark, but not too bad. I haven't replaced it yet.
The spark arrestor was well carboned; cleaning helped some.
The carb bowl had a little sediment, but I've seen worse. I cleaned it, the jet, and venturi tube with no apparent effect. The owner's manual does not give any info on carb adjustments.
Fuel pump seems to be working, at least at pull cranking speed.
Air cleaner element looks fine; no significant difference running without it.
Not done: Replace plug, adjust valves, compression check.
Currently I'm thinking that the lack of full power and the inability to pull even a relatively small load up quickly are the relevant symptoms.
I don't think there's a problem with the inverter controls. The control may be shutting the engine down when it can't keep up with demand, rather than allowing it a chance to catch up, which is how it's intended to operate. The inverter wants to supply either good power, or none at all.
My short term goal is to try to decide whether it's worth spending $50 on a service manual, or to just use the thing. The intended use is to run a couple lights and perhaps a laptop or TV for short power outages, so it'd probably do that as-is.