I've climbed snow-covered mountain roads (more like trails really. unpaved roads in a state park) in upstate New York in a Honda Accord plenty of times (friend has a rather remote cabin...).
Some people do legitimately have a need for larger vehicles. The farmer I use to work for had 4 or 5 F-150s to be used on the farms (no way you are getting a regular car in and out of some of those waterlogged fields). Those were farm equipment, not something you use to drop off the kids at school. People who need vehicles like that to go grocery shopping are an extreme minority.
No, it isn't. There is absolutely no need to have a truck-based, tall monster of a vehicle in even the most serious of winter conditions in the United States. 1: places will close of road conditions are that bad. 2: a front wheel drive with good snow tires is plenty good. 3: if you're that scared of driving in the snow, a subaru impreza with snow tires is small and does quite well in the most extreme stuff.
Although I replied earlier that I've driven a small sports car through most of the last three or so winters, what you said is not always correct. There are many times that the snow is simply too deep for cars, even that Impreza. When you're high centered on snow, you're not going anywhere.
And I have never seen any of the roads around here closed, even when impassable by anything short of a snowmobile or a tractor.
People in wintery climates who want SUVs want them because the roads are frequently uncleared, or because they need to travel at night during a blizzard when the snow is piling high. If snow and ice removal operated 24/7, I agree that a smaller car would be just fine.
As people will surely pile on to tell you, the vast majority of trips taken are no more than a few miles.
If U.S. cities had proper cycling infrastructure, I would expect many more adults would use bikes for local travel. But as it is, driving is certainly more reasonable for many people when the alternative is biking on a shoulderless road with deadly vehicles passing inches away from you.
If it was feasible and I lived within 10 miles of where I work, I would for sure bike. Right now, I commute almost 20 minutes in my car, so biking to and from work just isn't a great idea. If I lived and worked in the city, I would for sure bike every day. Even if I worked in the city, I would utilize the light rail for the majority of my commute. It sucks I would still have to drive 10-15 minutes to get to the light rail though.