The chief problems with going to a car wash are: a) they don't do a thorough job, b) you can purchase a lot of really good car care products with the money you've just spent, c) waiting in line is not productive, and, d) it starts snowing just as you leave the car wash, and you end up right behind the salting truck.
I go to the 8-quarter car wash in the winter because I can't use my hose in freezing temperatures. I always do my washing on a sunny day and after I pull out of the stall I hope the water on the side facing the sun won't freeze up while I'm drying the other side. If it does freeze, I just leave it because I'm worried about rubbing the ice and scratching the paint. At least it's clean ice.
Nobody told you that you have to live on the north pole! Every summer, I think that I could live further north but then every winter I complain that it is too cold. The few days we've had in the 70's has made me ready for summer. I also can't imagine driving on a salted road with a new car. I don't care how cheap it is, I wouldn't do it. I would have a summer car and a cheap beater for the winter.
Here in Toronto you are not allowed to wash your car in your driveway. Because the run-off is going down the rain water drain and eventually directly or indirectly goes into the lake. Not environmental friendly. I wash my car in self-serve car wash with my own soap with wax, and wax again using car detailing wax while wet.