I have killed a good number of deer with my old Mauser .35 Whelen and it's Lyman peep sight. If your eyes are sharp, then go for it. These days, I use optics on my rifles, as my eyesight is not as sharp as it was when I was younger, and find a low powered scope really helps, especially in the low light, thick woods I usually deer hunt in.
My last attempt at open sight deer hunting was with my Lyman black powder rifle that has peeps. Hunting a field edge in Tenn., a nice buck walked out at about 125 yards. The huge white front bead covered 3/4 of the deer, so I didn't shoot. My fault though, I should have replaced it with a smaller gold bead, but it had never been an issue with other deer, usually shot at 50 yards or so.
I still like aperture sights though, and use them on the range. Just bought a Browning model 52 with Lyman peeps - In good light, it puts them in there.
Larry