Ah, if only we lived in a more altruistic society. I went into transport for a career in the belief that it was all about finding ways for roads and railways to become more efficient, and also to encourage things like walking and cycling where reasonable. How wrong I was!
Part of the problem is that there's still too many engineers who think that roads are for cars (and not all of them particularly old either!), of course having too many people thinking that think that any mode of transport other than car is for the minority doesn't help.
Children should be taught that whilst cars are wonderful and have given us many great benefits, the opening of them isn't essential for the majority of people, nor should they be used by default when it comes to every trip. Unfortunately if a child is driven to school then they "learn" that going by car is the normal way of traveling and so that's what they want to do when they are older, even when walking/cycling/public transport may be more suitable.
Unfortunately government policy is contradictory, for example 2km is a suitable distance to walk to school, yet 3 miles (~5km) is when free school transport is provided. Which wouldn't be too bad of the 3 mile route had to be passable whilst pushing a cycle (so that for at least some of the route could be cycled), only the assessment of such walking routes assume that you can climb over gates, can pass through isolated woodland, doesn't consider that parents accompanying said child doesn't have other children with them (i.e. in a buggy) or to get to school (i.e. one at a primary school and one at a secondary school), not does it consider the fact that traffic had increased just a little bit since the routes were set.
I do think buses should, as many companies stop running all buses after a single snowflake settles.
Perhaps they have more grip due to their weight and thicker tyres, and they just stop running because the traffic builds up (see above re; other people need to have winter tyres too) but I am sure it would help. Buses seem to run fine in Sweden and other cold countries that get snow and ice.
I strongly dislike walking in icy conditions as pavements aren't cleared and when temperatures teeter around 0%, any soft snow soon melts and refreezes making it harder to walk than countries that stay well below zero.
In one country (I forget which) they found that it was better to grit the paths and leave the roads until later. The problem is the people setting policy are too car focused.