I hunt mostly red deer or fallow. It is often in thick bush. Sometimes in the open grassland type country. In the open country, then with open sights (or peep sights) its really not a big deal. 200 metres is as far as I have shot animals. I have not found these shots to be difficult at all. Have not needed to shoot further so far, but could probably go 250 metres, if I have a clear target, depends on the circumstances.

In the bush, at shorter ranges its actually a different story. The action is quicker, only a couple of steps and the deer is out of sight, and often the animals is half obscured by a bush, leaves, or a limb of a tree.
I used to use peep sighted rifles all the time, I have now gone to v rear sights with a bead foresight. I find it just as accurate. I have found that odd or inexplicable misses have dried up. Under pressure sometimes I am sure I was not centreing the bead properly. Despite what people say about it being automatic. But with a ghost ring, under pressure, I think I was actually ignoring the rear aperture completely sometimes, and probably unintended 'point' shooting. I think peep sights are good target sights. I am less convinced nowadays that they are the best iron rear sight for hunting.
I think there is a lot of rubbish talked about open and peep sights shooting.

One is about speed. The open sights or peep sight is much quicker than a scope. I know, I know, people are going to disagree with me and point to different tests done by various gun writers about how scopes are faster because of using the one focal plane and so forth....but they are wrong and I will tell you why.
If you put a scoped shooter up against an open sight shooter at a target. They will be the same, or the scoped shooter might well be a little bit quicker. This is target shooting where both shooters already know where the target is. It doesn't mean much. IF you have to aquire a new target each time - or if a single target is unexpected - the open sights guy will beat him.
Put them in a cluttered enviroment, with unexpected targets - never the same one over - at different ranges, and the open sights shooter will win every time, because if the shooter has to aquire a new target every time, the scope shooter will take more time finding each target in his tube. This is more like hunting, particularly hunting in bush, or 'thick or dark timber' as they say in the US.

I learned this shooting deer in thick bush at short ranges with scoped bolt actions, and with open sighted Winchester .30/30's.

The advantage of the scope, is precision in shot placement, because of the magnification of the target. This is offset though, particularly with offhand shooting, because of the wobbling of the sight, which causes shooters to snatch at the trigger. This is a distict definate thing, with open sights I am convinced people make a smoother trigger pull. Once again this relates to offhand shooting, which is my main stock in trade in bush hunting.
A real advantage of the scope is the ability to see 'through' the bush because of the single focal plane. This is important, a significant advantage.

Shoot through the bead. The point of impact should be in the centre of the bead. A six o'clock hold is for target shooting. When I sight rifles in I use a six o'clock hold - but I adjust the POI for the centre of the bead. Then when shooting on game it is instinctive to used the centre of the bead when under pressure, or I do, at least. Use a bead like a red dot sight. You shoot with both eyes open.

Springfield 1903 with wide open rear 'express 'sight and front bead - supposedly the most inaccurate type of open sight there is, or so we are told over and over again.
The three shots at right were at 50 metres, half an inch. The three shots at left were shot immediately after at 100 metres. Slightly over an inch. (White bull on black background is best for white or silver bead front sights. MAkes a big difference if your bead isn't bleeding into the white of the target paper...)

Fallow deer, open rear notch sight and front bead (Brno 600 in .270) 90 metre shot offhand.

Red stag, open buckhorn rear sight and front bead. (Winchester 94 .30/30) 85 metres shot offhand

You need to do a lot of dry firing and get used to the notion. It takes imagination to shoot well with open sights if you grew up with a scope. Dry firing a lot helps, I am convinced.

I prefer smaller beads - like those on the Brno 600 - 601 rifles, or about what the old Winchester .30/30 would have. I file them backwards on a 45 degree angle so they catch light from above.
Large beads such as a those put on some modern rifles apparently so they are better in low light - are useless. Dayglow, or firesights or coloured plastic beads are abominations.

I use iron sights a lot because they work for what I do, because the rifles are lighter and easier to carry and I have to travel long distances in rough country, and because I enjoy it, and because I know I am not really giving up much.
But if I was mostly shooting my deer in twilight from a stand, then I would have only scopes on my rifles.