If I’m not mistaken, dogs (like all other land mammals bar humans) don’t store fat subcutaneously as an insulator so they’re totally dependant on their fur plus their mass to surface area ratio and metabolic rate to stay warm. And if, like my mother, you’re curiously fastidious at keeping your dog shiny and white (he practically has his own bathroom…) then their fur loses most of its water-repellency so without a coat they’d freeze on a cold wet day.
Horses wear coats too, even though you see tiny ponies in Shetland half covered in snow and not in the least bit bothered by it. But they have a natural winter coat which hasn’t been clipped.