I am aware of the San people of the Kalahari (the articles i have linked to mentions them) and tales of their Persistence Hunting technique since they are the poster child for this myth. You should also checkout the various relevant papers linked to from my article links.
Regarding any "evidence" we have to be careful that it is scientifically valid i.e. done in presence of a impartial observer taking into account a) the terrain b) any support for food/water c) whether it is the norm in the group or not d) whether it was staged etc.
Thus saying "They have hunted like this for countless generations" is myth and not evidence. The "9 documented Kalahari hunts" were "9 reports" based on which the authors wrote the paper and are not a large enough dataset. If true, It almost seems like it was done on demand and not that it is the norm amongst them. It also requires a flat savannah type terrain like the Kalahari. All "video evidence" are suspect unless proven otherwise.
Read the articles i had linked to earlier and then these links/papers (linked to from the above articles)
1) https://www.sciencedirect.com/science/article/abs/pii/S00472... by Bunn & Pickering -
However, our understanding of the paleoenvironment, paleoecology, and archaeology of early Homo sites, reviewed here, makes us dubious about their further suggestion that ER might have been employed regularly and successfully in service of that foraging pursuit. We are reluctant to assign to early Homo...
2) https://en.wikipedia.org/wiki/Endurance_running_hypothesis#A... -
Small changes in joint morphology may indicate neutral evolutionary processes rather than selection ... Most of those proposed traits have not been tested for their effect on walking and running efficiency ... Relative lower-limb length has a slightly larger effect on the economy of walking than running. The heel-down foot posture makes walking economical but does not benefit running ... hindlimb proposed traits were already present in Australopithecus or early Homo. Those hindlimb characteristics most likely evolved to improve walking efficiency with improved running as a by-product ... The data suggests that the large size of the gluteus maximus reflects multiple roles during rapid and powerful movements rather than a specific adaptation to submaximal endurance running.
3) One early famous film The Hunters (1958) which depicted the "persistence hunt" of a giraffe was later admitted to be a fake. From https://afan.ottenheimer.com/articles/myth_of_persistent_hun...
What did he admit to?
That the film was shot over years with different San, with different giraffes, that the poisoned arrow was shot from a moving jeep, the San were riding in the filmmaker's jeep chasing the giraffe, and that the San returned to base camp each night because "without the reassurance of the supplies in his Jeep, the Ju/’hoansi might not have been prepared to pursue the giraffe for as long as five days since they could have died of thirst out in the middle of the Kalahari." Some have also suspected that the giraffe was taking so long to die that it was shot with a gun to get that final dramatic collapse on film. But, by the time these issues were admitted ... the film had ingrained itself into the mindset of the public having "learned" that early humans' evolution of endurance helped them to become top of the food chain.
Later it was found that MANY infotainment nature films did similar things because they were more interested in captivating the audience than telling an honest story.
Also from the above link;
To Summarize: The persistent hunting theory would need to have the following evidence to support the theory.
1. The fossil records should show humans' tool damage on prey fossils before animal damage on those same fossils.
2. A review of other hunted animals that supports this "sweat vs panting" theory.
3. An analysis of human energy expenditure vs prey animal energy expenditure that shows humans are better in long distance running/endurance.
4. An scientific experiment showing a human successfully doing a persistent hunt to chase down an animal like this to exhaustion with nothing but stone-age tools.
Unfortunately for the persistent hunting theory and Lieberman's paper:
1. None of this evidence was presented in these papers.
2. Subsequent studies have shown that each one of these claims fails.
For each of the above points the author provides links and details (under "The Evidence" section) which you can checkout.