I bike year-round with my 2 kids (and sometimes a dog too) in Denver, Colorado. Having the right setup has made it possible for me to be car-free with kids, yes we do have a minivan that my wife drives but there are many times that having that available simply isn't an option. Over the past few years I have pieced together a winter family biking setup that I think could work until your kids are 5-6 in a pinch (like they are really squeezed into the trailer)!
A Kid Bike Trailer
The most important piece of this is the trailer because I know that if I slip on ice or snow, they are safer in the trailer with 5-point harnesses and a roll bar system. Also, the hitch on a burley trailer does pivot so you have some give if your bike was to tip over that it doesn't tip the trailer over.
Fat Bike Tires on the Kid Trailer
A few years ago Burley came out with their 16+ wheel kit which is basically fat bike tires on a trailer. I use these and not the ski attachment because in Denver the snow melts and the bike paths are plowed. They make it so I'm not dragging the trailer through the snow and instead pack down the snow.
I will say that there are many days that I use the above setup (trailer and 16+ tires) with my Urban Arrow cargo bike but on days where there is 4-6+ inches of fresh snow, I pull out the fat bike plus after school drop off I take the path less traveled and get in a little snow fat bike riding on the way to the shop!