In Born to Run McDougal references The Art of Tracking: The Origin of Science by Louis Liebenberg as evidence that humans have been persistence hunting in Africa since forever. I have only read bits of it, but it doesn't really say that.
Mostly it's about how important it is to know what an animal will do from the signs it leaves behind.
All the methods would take endurance. Even ambush hunting takes legwork to know where the herds will be as they move. But the important part is the brain work, not the ability to persist through stupidity.
From Chapter 5: Hunter-Gatherer Subsistence.
Occasionally, small animals may be knocked down with a throwing club and finished off at close quarters, or if the animal is stunned and takes off, it may be run down. Large birds may also be knocked down with throwing clubs. The young of small mammal species are frequently run down on foot and caught by hand (Lee, 1979). Slow-moving animals, such as antbears and porcupines, are easily run down when encountered in open country (Silberbauer, 1981). Animals such as eland, kudu, gemsbok, hartebeest, duiker, steenbok, cheetah, caracal and African wild cat may be run down in the hotter part of the day and killed when they are exhausted. The animal is stalked and startled to make it run while the hunter follows at a steady pace. This process is repeated until the animal is exhausted and can be finished off with a spear or club (Steyn, 1984a).
!Xo hunters at Lone Tree Borehole, for example, use this method, and concentrate on different species at different times of the year. Steenbok, duiker and gemsbok are run down in the rainy season, because the wet sand forces open their hoofs, thereby stiffening the joints. Kudu, eland and red hartebeest are run down in the dry season. because they tire more easily on loose sand.
In the early summer, before the rains break, animals are poorly nourished. If a ruminant is prevented from chewing its cud on the chase, it develops indigestion which eventually slows it down. This enables the hunters to come close enough to kill it with spears (Heinz, 1978b).
In woodland, where visibility is limited by the vegetation, the animals may run out of sight and hunters must track them down before they have a chance to get enough rest. When running down a herd of kudu, for example, trackers will look to either side of the trail to see if one of the animals has broken away from the rest of the herd. They will then follow the animal that broke away. When it starts to tire, the weakest animal usually breaks away from the herd, to hide in the bush, while the others continue to flee. (Since a predator will probably follow the scent of the herd, the stronger animals have a better chance of outrunning it, while the weaker animals have a chance to escape unnoticed from where they have hidden themselves.)
The success of this method depends on how quickly the animal can be tracked down. The most important factors are the hunter's tracking abilities and how difficult, or easy, the terrain is for tracking. In the immediate vicinity of Lone Tree Borehole the grass has been heavily overgrazed by cattle and the ground is quite barren, so it is relatively easy to follow spoor in the sand. The woodland, on the other hand, is still adequately vegetated for browsers like kudu. Further away from the borehole, where the ground is less barren, it becomes more difficult to track down animals quickly, while in areas where the ground is hard it would be very difficult to track fast enough to exhaust the animals. In difficult terrain the chances of success are slender unless the animal is weakened by injury, illness, or hunger and thirst.
That's one page about persistence hunting, in the 16 page chapter of hunting techniques of the Kalahari. Stealing kills from predators, ambushes, and cooperative ambushes are what he thinks early hominids did, but they take up less space.
The other 170 pages of the book are about the brain work, not the leg work.