So here's some ideas to help solve that potential problem:
1) Make all clothing bigger.
When I played H&H I rarely carried many sets with me to begin with. If I needed something different to wear, I would get it from my wardrobe or better yet a chest nearby the activity I wanted to do (cooking set in the kitchen for example). But this the wilderness, far away from your base. By making all clothes bigger, players would need to manage their limited space more for when traveling to harsher climates. I would assume that all clothing would offer SOME protection, even if marginal, but I always found it a little funny how a pair of leather pants can be the same size as both a pair of linen pants and shirt.
2) Exposure has a timer.
Like in Breath of the Wild, and Ocarina of Time, exposure to the elements doesn't instantly kill you, the amount of time it would take to kill/start to kill you is about 10-20 seconds. By equipping gear you can extend your time. The problem with this is it would prevent living on the mountains without a constant source of fire / or a house/cave. Yea people will still carry with them the clothing they need, but they wouldn't be able to stay out in the cold forever.
- This would also assume you can't just keep putting on a different coat every 5 minutes and be in the clear. If your bear cloak gives you +10 minutes and when you have 1 minute left, putting a new bear cloak won't give you another free +10 minutes.

3) Gear needs to be repaired (for exposure purposes only).
As you travel through the snow, your clothes start to lose their usefulness. A strap comes undone, the fur is covered in ice, you tripped and now a sleeve is ripped. Periodic repairing being necessary can help keep players from trying to over extend their welcome.
4) Tea VS Beer
You should really be drinking something warm over generic water or booze to stay warm. Have warm tea extend your timer in cold climates. Also have the waterflask keep the warm tea....err... warm longer.
So what ideas do you guys have that would help prevent exposure from simply becoming a "not-problem"? Frankly I think a timer of sorts is all that's really needed (mainly because a harsh winter would cause everyone to always wear the needed clothes). A big difference between BoTW and H&H is inventory space, where BoTW has almost an infinite inventory for clothing and food, H&H is rather limited.