Thin tires may not be as much of a problem as you think - I've been riding through the winter for several years on a 90's mountain style bike using pretty thin tires - 26 x 1.50 - and they have much better traction than you might expect in snow and on ice. I do ride defensively if the roads aren't clear though - a little slower, and if there's a lot of snow I do try to avoid heavy traffic roads. That's more due to lack of confidence in the automobile traffic, plus splashing slush if its warmer, than worry about crashing.
I see a lot of folks say they have luck with the skinny tires in winter but they have never worked well for me. I use 1.9 to 2" mud tires. They are big enough that they don't get too squishy in the biscuit dough, and they still sink enough to get traction on the slush over the ice. I fell a couple of times on ice-under biscuit dough a few years ago and so I now use 2" studded tires and won't look back. They are pricy though, so don't buy them until you know you will use them.