That may be true up North but not down here in the Deep South. It will never be that cold.
If what you say is true, all our meat would be awful. It isn’t.
Still, in the older times that once were, most butchering happened in the late Fall. I think though that was because there’s less green forage in the Winter and providing grain and hay would cost folks too much.
As far as game goes, it’s deer or turkey and over the past few decades, wild hogs. Deer are shot in the annual early Winter rut, and turkeys in the Spring. Hogs are shot year round.
Get the carcass to the locker.