Have killed quite a bit of big game with "iron" sights out to 250, and even one bull caribou (about the size of a big mule deer) at 350, using a pre-'64 Model 70 .270 with a Lyman aperture rear and the factory front bead. Have also used 2-1/2x and 3x scopes out to 300+.
The big advantage I've found in more magnification is (as others have observed) seeing antlers in dimmer light, whether just to discern if the deer's legal, or whatever. Around 35 years ago I hunted a lot with a K3 Weaver, which was old enough to have single-coated lenses, or no coatings at all. Was also more addicted to "trophy" hunting than I am now, and one morning a small herd of whitetails wandered within 200 yards of where I sat along a creekbottom fenceline. Through my fine Bausch & Lomb binoculars I could see two were bucks, a young one and one with much larger antlers. But when I aimed with the K3, I couldn't see antlers on any of 'em!
It took some fast switching between binocular and scope to make sure exactly which deer was the mature buck, and I made the shot. But wasn't really sure I'd shot the "right" deer until walking up to where he lay.
