@ Bobbo and Dutch, I understand the concept, but real world, the concern is virtually none.
Examples, just the recent ones with my own generators, 7kw backup power for the house. It's probably 15 years old? Runs when the power goes out. Sometimes, several times a year, during the rainy periods. Sometimes sits for a year or more.
Honda 2k, hadn't used it in like 2 years and used only during summer trips occasionally. Just fired it up a couple weeks ago to use it briefly. No issue.
Our "new" camper, 4+ years old. Onan genny had a total of 8 hours on it as of this spring when we bought it. Owner hadn't camped in it or used the generator since 2018. Somehow it fired up and ran ok on the 4 year old gas in the tank...and no issues with the generator, generating.
Is it better to use it than let it sit indefinitely? Absolutely, I believe so. Not unlike any other ICE powered anything.
But if one takes the right care (draining fuel or treating fuel, run carb dry if possible, etc) then no harm will come to the generator by sitting over the winter.
Also, OP is in NY. It may not be a good idea to heat up the camper in the middle of winter. Or may be more effort than it's worth. If you heat up a camper covered in snow, once you start melting the snow, you kinda need to get rid of all of it, so it doesn't turn to ice.