For "just hunting", I like a long, medium weight barrel--but I hunt in country where long shots are probable necessities. I handload, and go for max velocity.
If I'm mostly going to shoot paper, a heavy barrel when set up properly is the most stable for long strings of fire. That is, something like, say, ten shots in a limited allowable time period. This sort of rifle is also desirable for shooting prairie dogs, where you have numerous targets and relatively short intervals between shots.
Basic coyote and deer hunting, lighter works as well as anything, really. What's important, IMO, is that the first shot from a cold barrel always goes to the same place today as it did last week or last month. Few rifles won't keep the first two or three shots closely grouped, even those with thin barrels.