That's a very effective "in range" strategy for trapping, if you have the foresight to bring everything needed. You can catch elk and reindeer the same way if you bring a couple turnips for bait. The bear trap or trap-pit will catch those big elks for you too. Berries if you're catching birds.
I caught some runaway foxes this way with no trap at all, by setting up a piece of meat when I thought I was in range of the hiding fox, and then hiding myself a good distance away with a loaded crossbow. With some luck and patience (passing time with the "." key held down till you see movement) the foxes do come to sniff the meat after they calm down from the chase you gave them before, so you can get that perfect straight line shot.
Btw if you ever wondered like me, what happens to animals when you leave the area - I can confirm that they don't just despawn. I went seal hunting recently near the sea area and been peppering this one poor seal with arrows over 4 days in roughly the same neighbourhood where I first found him. I felt like giving up several times, thinking for sure he just despawned and it's fruitless. After many days, it was a triumph to finally down him with a lucky shot - I got a seal that had 8 different arrows in him, so for sure it was the same one! Each time I'd hit him, he would dive and then I could only rarely find him the same day a secodn time, but I'd go back to camp and sleep, then return when it was bright morningtime again and start looking for him all over again. I was finding that seal on zoomed-in map mode, even though a lot of the time he could not be seen when zoomed out.
So my theory is that animals won't just de-spawn, they may wander off though. But they're still there on the global map, and there's a chance you might find your wounded elk some day still even if they get away on you in the chase the first time.