Terrain and how hard the critters have been hunted in prior week will influence how easy it is to get within 200 yards. Presumably Thag and his spear had a big herd to follow and focus on the sick and young. Of course, buffalo jump cliff strategy would work quite well where animals are spooked towards a cliff and they fall 20 feet or more. Not sure how to call that kill distance.
Hunting water in a dry year is ethical but if a critter needs to come to water and you are in a blind at the only water holes for a mile in any direction then you should be able to get a close shot.
I have seen a spooked herd of pronghorn run 50 yards and turn to look back much like a whitetail deer will. I have seen herds run no less than a mile before stopping to turn and look back. Hunting pressure and terrain are the difference, in my opinion. Your skill to close the distance on a spooked herd on flat terrain may be great. I have tried crawling and duck walking to close the distance on flat terrain and a walking herd browsing on sage can outpace my efforts.