Does injury not related to the legs still slow/tire animals?

[Willy]

I tried hunting an elk with a bow but only managed to get a shot in its body before it took off. So I decided I would try running down the elk til it was breathless and it work like a charm (the elk wasn't leaving blood trails so I dont think it was bleeding). But then later that character died to a bear ambush...

But I really liked the idea of endurance hunting so on my new character I got a spear and carried nothing but shoes and some pants. But I can not seem to endurance hunt to safe my life. I chase them for ages and when I examine them there is no status saying they are tired or anything (is that related to eyesight or something) and they managed to slip away.

[PALU]

I believe players are slowed due to non leg injuries, so animals should be as well. However, the key to endurance hunting is to stress the animal so it runs (to build up fatigue), and do so often enough that it doesn't have time to recover in between being forced to run. At the same time, the player can't run constantly as that will slow the player down to a crawl.
The speed of the player is important, I think. A high speed player with a 7 km/h walking rate seems to be able to get elks and reindeer to run into the ground by walking (provided the tracking is sufficient), while my current 5 km/h player (about average) was thwarted by a reindeer who could recover by just walking away in more or less a straight line.
If you can corner the prey so it constantly runs back and forth along a river or lake shore it's fairly easy. If you can drive a calf away from its mother is should be reasonably easy as well, as the calf should try to return towards the mother, and if your player is in between to cause it to run again you may be able to bring it down reasonably quickly, and the same goes for reindeer in a herd, as they try to return to the herd, rather than walk away from you in a straight line.

I don't think eye sight affects your assessment of the prey's status: I think you get it all if you're able to see it (and I think eye sight only affects detection).

I'd suggest you take another look at your new character's speed stat (or just look a the walking speed), as I suspect a 3 km/h slowpoke would have trouble endurance hunting anything not hemmed in. If the character has poor speed, you may consider changing the strategy (e.g. towards traps).

Terrain matters as well, of course. Spruce infested forest is a pain due to the poor visibility, so you're often reduced to tracking rather than moving directly towards the target when it comes into what should be visual range again, giving it more time to recover before you scare it into running again.

[mosshobo]

It do slow them down! Depending on how much damage taken and on what parts of the body they are hurt on they can limp, get crippled and so on.
If you hurt one or more legs badly they cannot even run away faster than you can follow and you can club them down.

One minor puncture is not enough to slow them a lot, so they can be hurt on their legs and still outrun you with ease.
You need to get a good hit or multiple bad ones.

It's a very good game mechanic imo.
I had a limping badger that was frequenting my camp back and forth for over a week until I finally managed to get another shot on his leg and made him crippled so I could get to him.

Edit. I did not read your question properly.
I think other injuries can slow animals too and make them get tired faster.
I shot a reindeer in the guts 2 times and it was bleeding, it seemed it moved a little slower than the other reindeer in the group.  I maybe shot him in the leg too, don't remember. But could eventually kill him with an axe.