Finally! The EU2000i is running properly – like new.
Had to clean the carburetor one more time and that did the trick.
For posterity, this is my best guess summary of what happened to my EU2000:
After 6 years of exposure to gas (perhaps ethanol an aggravating factor) the fuel line right after the fuel filter went bad (brittle and cracked) deformed and obstructed the flow of gas out of the fuel filter. Also, the once white fuel filter turned yellow brittle.
After re-assembling, having previously added Techron to gas and having sprayed carb with Carburetor Cleaner some more accumulated gunk got loose and carb soon got contaminated once again.
A final carb disassembly and cleaning (very thin wire plus carb cleaner) and new gas finally set things straight.
Finally, here is a picture of the degraded fuel line (I sliced it to show the degradation) :
The interior had completely degraded, the thin exterior sleeve (apparently a different material?) was holding the whole thing together, but the deformation was apparently blocking the fuel filter exit and thus no fuel was reaching the carburetor.
All is in working order now.
Thanks everybody for all your help.