TL;DR: don’t put a scarf over your nose in very cold temperatures.
TL;DR: don’t put a scarf over your nose in very cold temperatures.
I've never had a problem with a scarf freezing stiff like you describe. They build up ice crystals of course, but they don't interfere with air flow and are easily broken.
The biggest problem I expect from wearing a mask in the winter is that they'll get damp quickly because the cold temperatures, and perhaps having a scarf over top, will prevent moisture from evaporating.
In such cold climates I wear an ultra long scarf (2m-2.5m), I wrap my neck/jaw/ears 3-4-5 times (depending how tight) and I tuck it in my jacket. The outside of the scarf freezes, the inside remains warm. I also try to breathe by my nose, less humid.
Weird, as a Canadian who has been through many -35 days, I absolutely use a face covering, whether that's a scarf, neck gaiter, mask...Never had any problem like that.