Thank you all for the great replies, and a really useful discussion. Apparently, one cannot go wrong with the EU Hondas.
Regarding my Yamaha EF2400iS, I found a local Yamaha MotorSports shop that said they work on the inverter-gens, so I took it over there Saturday. Told him all the symptoms, and he just nodded and said it's a pretty easy service fix in the carburetor and fuel path. Sounded good anyway, and hopefully he can breath some new life in the 10 year old EF2400iS.
If not, I'm definitely going to get a Honda...
Cheers!
- Dan