I think you might be stuck with the school pick up. It's for the very reasons you have concerns that I just don't take school age kids any more.

When I worked in daycare, the cold weather policy (according to Day Nurseries Act) was that kids were kept inside in any weather that dips below -10 degrees (including wind chill). Personally, with little ones, I find that even too cold. I won't take kids out to play if it goes below -5 degrees and I'm very honest about that at intake interview. Especially if the children are very young, it's just miserable for them as they aren't as mobile as an older child. Not being able to move around well in their snowsuits definitely hampers their ability to generate body heat, so they get pretty cold pretty fast. Sooooo not worth it!

For hot weather, I just use my best judgement. If you open the door and are hit with a 'wall' of heat that sucks the breath out of you, don't go out. We'll go out in hotter than 32 degrees depending on what our activity is. If we're playing in the sprinkler that day, in my experience, it's been fine. For regular play with no heat relief, it's pretty miserable.