Those things are only 10 - 12" in diameter. Pizza peels aren't much thicker, maybe 5/8", but are much larger and they stay pretty flat, certainly no large cupping. I don't think it mattered that old homes had no central heat, they still got dry as a bone in Winter, at least in the kitchen where the cook stove was and where these things would have been used. I don't think QS stock is key either. Plain sawn stock 3 or 4 inches wide with every board flipped should be fine, again, like a pizza peel. Just don't glue in the batten. Set it in a shallow DT and pin the center.
John