Tbh I always forgets about the snow but that's a problem other cities have dealt with. In particular we should look at the Netherlands, where 75 percent of children below the age of 12 bike to school (many with parents to be clear especially the younger ones).

In the Netherlands this article talks about snow. Essentially they have a prepared winter maintenance network to clean the bike lanes quickly and in order of priority, somewhat similar in idea I think to the winter street network we already have here to clean the roads since that horrible winter a few years ago where everything was snowed in for like a week.

As for ages, in the Netherlands many children bike from ages 6-9 with their parents, and after that often alone. A strange idea I know in the US where children are rarely let out alone, but that's not actually due to the danger of strangers, but rather due to the rise of suburbia and car culture designed cities https://youtu.be/oHlpmxLTxpw . Which is also why NYC is a bit of an exception to this already with arguably the only real public transport service.

https://www.hollandbiketours.com/news/how-do-dutch-children-learn-to-ride-their-bikes-in-traffic/

https://mobycon.com/updates/the-five-pillars-of-dutch-children-cycling/

https://www.linkedin.com/pulse/cycling-school-netherlands-numbers-what-can-we-learn-%C3%B5nne-kask#:~:text=It%20was%20normal%20that%20children,child%20alone%20in%20the%20traffic

>when having to travel more than a couple of miles, especially in the winter

First of all students should not have to travel miles to get to school within nyc. The city is dense there should be at least one school within a mile of people if not multiple.

Admittedly most students who use the bus are going to a school of preferred choice in another neighborhood but even taking that into account the average school bus commute is 20 minutes in NYC. Bikes are signficantly faster than buses with (less stops plus speed limits plus acceleration) so that's only 10 minutes via bike give or take 3~4 minutes depending on the bike and assuming good bike lanes.

Second of all you could even add in Ebikes. Ebikes are incredible, they require far less effort to peddle and are also faster, although I would stick with regular bikes since it's good exercise. I recently tried one for the first time (the nice white citi bike one) and rode in a circle around the entire Central Park easily and I'm out of shape. They are straight up just better than cars except in terms of safety when on car dominated roads.

Really the biggest concerns are the demographics of students using buses. Just under half are homeless. I'm not sure whether it makes sense to offer them bikes as an alternative. It seems to me it'd be superior since they get a free bike to transport them around in general. On the other hand they likely might not have the space or security to keep it. So I'll assume it wouldnt work for most of them although I'd guess at least some could.

Plus amongst the remaining half a significant number take the bus due to disabilities. Obviously these disabilities would prevent biking. Although I imagine there could be special ebikes for some disabled people.

TLDR: bike paths can be maintained in winter, older children (above 9) can bike on their own if we create better bike lanes with a more expansive network, and bikes can easily transport students as far as a bus would take them within nyc.

However after reading about bus demographics in NYC it seems bikes don't make sense for many if not the majority of the students who use school buses anyways.