Two of the most cycling friendly cities in the world are also cities with a lot of rainy days (Amsterdan and Copenhagen). A lot of people with kids there will disagree with you.
It seems to me the biggest issue people who never cycle think stops people from cycling is weather, while in reality it is unsafe infrastructure. Build the infrastructure, people will cycle and will not bother about the weather.
Overall I think it's a cultural/convenience thing.
also as an aside there's a lot more potential for variance in travel time on a bike and if a school has punitive tardiness systems, this will get taken out on the student a lot.
I can't think of many situations where schedules collided to the point where I needed to take several bulky items home and back on the same day. And worst case, I'd just leave my sports kit at school overnight perhaps? Or get a lift on that one day. Or take the bus!
I can't speak to large instruments though, that's a bigger (although fairly niche!?) problem.