Under some conditions. What if you're 50 yards away? Which would be your preference, a 180 grain bullet out of a .30-06, or a load of #7s?
But the main difference is, if you're bird hunting with an unloaded shotgun and a bird flushes, you're not getting a shot. So if you bird hunt with an empty chamber you may as well stay home.
Not true with big game and a rifle. Not all big game hunters are deer hunters in the woods of New York, jumping deer and taking shots at running deer.
By far most hunting accients, take place at ranges measured in feet, not yards. I haven't shot at a running deer in many years. If you're a "stillhunter" and you're jumping deer and taking running shots with any regularity, you're doing it wrong, plain and simple.
