John's "C" type comes closest to the old Winchester 1895s I've hunted with.
I've taken moose up to approx. 70 metres, the biggest one I shot I measured to be @ 50 metres.
Typically, Finnish moose hunting is close quarters job so a scope can be a hindrance due to the limited field of view. In the thick woods the open sight is useful, indeed.

I can hit a 3" bull @ 150 metres on bench rest with my new Winchester. It has Recknagel sights whereby the V is ve-e-e-ry wide, it has a small dot at the bottom and the front sight is a small bead. Very accurate on the range but on game I always used the red dot sight, however.

If the game is moose size I'm easily comfortable up to 100 metres, maybe 150 (depending on whether I have support!).

As Marrakai said, it also depends a great deal on the type of the sight. Older rifles tend to have better open sights, as they were the norm and thus they had to be good. Today, they tend to be auxiliaries or meant for short distance driven hunt ("Battue" type sights).
I'd say that if the game is smaller than moose, the max is about 100 metres and for moose or bigger, up to 150 metres.
If the light is good and the user is familiar with the sights, then this can be upped.

I noticed that the red dot sight covers too much of a typical safari animal at 100 metres to be useful. It's close-quarters fast shooting sight for sure; despite being quite accurate in bench rest shooting.
This also applies to the open sights: if the front sight is a big bead or a wide post, it covers too much of the animal to be accurate. Smaller sights suit long distances and smaller game better.

- Lars/Finland