I live in Chicago and my family and our neighbors transport our children to school/stores/activities via bikes or feet. When the snow gets real bad we switch to sleds.
Anecdotally from observing my kids school I’d say that yuba bike linked is very popular. As are the Dutch bikes with the big front thing for hauling kids.
That of course relies on living in a place with infrastructure that enables it. Which is what Berlin is trying to build...
Plenty of people in northern Scandinavia cycle throughout the winter. With the right clothing and equipment, the weather isn't a particular problem. Many of the highest-rated cities on the Copenhagenize Index of cycle-friendly cities are far north. The highest-rated city in the US is Minneapolis. Malmö, Helsinki and Montréal also make the top 20.
People from non-cycling cultures tend to badly under-estimate the quality of utility cycling equipment and over-estimate the impact of weather on a well-equipped rider.
Schwalbe Marathon Winter tires have hundreds of steel spikes and will grip securely even on black ice. SKS Longboard fenders provide a huge amount of protection from rain, spray and road grime. Lobster mittens keep your hands toasty warm in arctic conditions, while still allowing you to operate the brake and gear levers. Epicyclic gearhubs provide up to 14 gear ratios in a completely enclosed and practically maintenance-free package. Good LED bicycle lights are stunningly bright, rivalling the performance of car headlamps. A long john or Christiania cargo bike will comfortably carry two children, a week's worth of groceries or even a washing machine; with electric assist, these bikes are viable even in very hilly areas.
As an aside, many of the LED bike headlamps I have seen (just like LED car headlamps and most LED hiking/camping lights) are unpleasant and a safety hazard. They are very bright in the blue part of the spectrum, which means they cause severely distracting/blinding glare for anyone else on the road, and knock out people's night vision. Personally I think they should be heavily regulated and forced to be less bright and maybe larger or more diffuse, with a more night-appropriate spectral power distribution.
I have a similar bike in Boston with two kids. It's not that bad; snowpants, good gloves, and adult wind breakers thrown around them gets you pretty far. The hardest part was getting the 2yo to wear a neck warmer or face mask when it was in the 20s. I imagine a trailer would be warmer and safer on ice. The long tail is just nicer overall.
