Yup... its the motor starting loads that are the killers to these generators. Even alot of the really large gen sets i set up on community well systems require a soft start (frequency drive) to gently ramp up the motor.. otherwise the generator trips off.
I have a 4,500 +/- SF house on a 10K generator. All major appliances are gas. My generator panel has twin guages showing loads on each leg. There never more than 25% of peak. And im running all lights, 1/2 of kitchen, freezer, built in refrigerator, furnace (110 fan), twin water heaters (again just exhause 110 v fans and circ pump), tv, bedrooms, bathrooms and garage and garage door openers. Oh...and christmas lights (to piss off the neighbors).
I prob could get along fine with a 5kw system. The a/c is a heat pump, so thats out no mater which system i use, but im in N. Idaho so no issue. Heat has a gas emergency backup.