Other than pressure washers I have never run into something that said start it and use it right away, idling was bad for it. Might be common or just the one my uncle borrowed, but it said to do that so we did that.
Letting the generator get somewhat warmed up a bit before placing a load on it might help, some generators might care more than others.
If I had it in a garage I could easily see using a kerosene heater, I have a bunch of em, in the garage so it is warmed up and easier to start even if that meant starting it at the garage door and then moving it outside. I have smaller stuff that fits on 4 wheel dollies and what not.
This is more for pull start stuff that is not starting easy. If you are one of those who can get pull start stuff to start easy peasy then don't worry about it.
I generally wind up with the pull cord breaking or me getting annoyed before it fires in really cold weather.
A vehicle is designed with a whole lot more than some of these cheap generators.
I have no clue about the snow mobile comparison, never had one. I know carbed atv's can have issues if they are not cared for but then that is usually what we come down to, is it cared for and known what is needed to start in the cold or is it only used on pretty days when the temps are 60 degrees above zero.