Open sights are quite different than peep sights. I haven't hunted with open sights since I stopped hunting with my Dad's or grandfather's old 241 Remingtons in .22 Long Rifle.
Peep sights are a much different matter. My first big game rifle was a Marlin 336 in .35 Remington equipped with a peep sight.
Since then I've hunted deer, elk and alot of small game with peep sighted center fires. Rifles included two Remington 760's, one an '06 and the other a .257 Roberts, a .25-20 1894CL Marlin, and a Model 39 Marlin .22.
Scopes help in almost any hunting situation. They allow one to see better. Since game has a bad habit of not standing in well lit, open situations, a scope can help alot.
I haven't found peep sighted rifles to be much, if any a drawback at the shorter ranges, say 75 yds. and under. As far as shooting well enough, even at ranges like 200 yds., my iron sighted guns shoot very acceptable groups. At present, I'm planning to do more practicing at even longer ranges with the 760 Roberts. Might even do another big game rifle in a peep sight if that work well enough.
You might consider buying Barsess's book, Obsessions of a Rifle Loony. Good comments of using peep sights for hunting.
One thing that has stood out about good peep sights is that they are far more rugged than any scope. If your hunting rifles get knocked around much, this is nice to have. E

Last edited by Oheremicus; 08/31/15.