I had a 3500w Champion Generator that travelled coast to coast with us in the back of my pickup truck (and rode there 12 months of the year, at that, for about 5 years) that we used when we had our 5th wheel. ALWAYS started first pull, would run the big roof mounted AC unit on the roof of the trailer as well as every other appliance, and never caused me an ounce of problems. Sold it 5 years later for $50 less than I paid for it. It's actually a pretty solid brand.
Problem is...it was an open frame single-speed (non inverter style) generator. Fine for running when noise wasn't a concern. Eventually upgraded to a Honda EU2000i inverter generator which was the cats meow (quiet, even under heavy loads, nearly silent from 50 feet under light load) but waaaay more expensive. But for us it was the difference of being able to run it in the non-electrical areas of campgrounds all while still having all the creature comforts in our RV, but (most importantly) not being "That guy" with a loud generator wrecking the serenity. I put it in the bed of the truck, pulled the toneau partly over it (damped noise, still allowed ventilation), and it while running at low RPM just charging my golf cart batteries, barely audible even on our own campsite.
Back on the topic of "powering your house during an ice storm", my inverter solution has kept us afloat (furnace, fridge, and electronics/TV) for 24 hours before, although I know it's not for everyone, and it's also not automatic by any means.
Fiddles, it seems what you're looking for is a fully automatic standby generator. Look online at Home Depot's website, they sell the Generac models - I actually deliver them to a lot of the stores.
The smaller (6000w) units can be had for <$3000, although you'll easily be in for probably another $2K for installation by the time you have a gasfitter run the gas lines and an electrician do the wiring. But it's going to do what you want it to do...as long as you can get it situated somewhere permanent where the gas and electrical hookups are workable given your housing situation. And 6000w is plenty for careful usage, even being enough to run an element on the stove or small appliances like a toaster oven so long as you're not trying to run everything that's high demand at the same time.