I also use hardware cloth covered with hay. I’m not all that cold either, it seldom gets much below zero Fahrenheit as an overnight low here. It’s normally a little warmer than that when they actually lay.
I’ve read that wire bottoms are not as hard as a solid wooden bottom so the eggs are less likely to break when laid. I’m not sure I totally believe that one. But the wire bottoms do help keep the nests cleaner, a lot of trash falls through if you stir up the nest. To me another advantage is in the summer. While the hay does block it a lot, the wire allows some air movement on those really days. You don’t want your nests to become ovens and cook the hens when they are laying.
My broody hens have no problems hatching eggs in those nests with the hay.
I'm near Prairie Grove so you know our climate.