what other animals? I used the dog example because it fits perfectly here: they don't sweat yet it's impossible to catch up to them if they are running away. Even if they overheat they don't just drop dead, they slow down to a human's walking pace.
And what about running animals that do sweat like horses? makes no sense. What makes more sense is to approach from multiple fronts and spear/arrow it down. Plus, all remaining hunter-gatherers hunt with the help of dogs. Sure, you do have to run, a lot. But the endurance side is the least important.