Back when white gas backpacking stoves were novel and only used by ”serious mountaineers” (ie: multiple days above tree line), us mere backpackers used camp fires to cook and melt water in the winter with no problem.
The secret is to split the wet/frozen wood (that was covered with snow) into pencil to thumb thick pieces, and use a cotton ball plus a hat full of shavings from the inside of one of the split pieces to get the fire started. 2 vaseline coated cotton balls per fire should ensure success even for damp wood as long as you make lots of shavings.
Wrist thick branches can be split with a sturdy hunting knife. Hatchets are heavy :) A folding pruning saw is great for removing limbs from downed trees when shoveling down to the ground to find wood is too much work.
Ps: Stoves are lots more convenient, and lots faster. But making a fire in rainy or snow covered conditions isn't rocket science ;)