Anecdotally, I am in Madison, WI and it would be feasible with kids in the right neighborhood here. Lots of bikes have little carriages for small children.
Madison WI is that way too. I'm a year-round bike commuter here. Studded tires keep me shiny-side-up.
But I'm torn about pushing winter cycling on people. I realize that if it's miserable for somebody, then they're not going to do it. I happen to love cycling and bikes, so the idea of riding during the winter just means more chances to ride under different conditions, and a chance to tinker with one more bike.
In terms of ditching the car altogether, that's a toss up. For most families, to understand their commuting situation, you have to draw the triangle from the two parents jobs to school or day care, possibly different schools if they have more than one kid. For a while, my daily commute was a labyrinth involving dropping people off at different locations before turning the minivan around and heading for work. But today, everybody takes a bus or rides their bikes. Our high school has basically no student parking. Cars are part of our transportation mix, but we try to minimize their use.
At one point it had been so long since I drove to work, that I forgot about a big construction project, and got lost.
They also make bikes with wide tires now that perform admirably in snow. With the right gear, it's not too bad unless you are trying to ride through a foot of snow.
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