Wikipedia politely labels persistence hunting as "conjecture". It's interesting that pretty much everything important from our ancestral past (e.g. fire-making, flint-napping tools, spears, skins and furs etc.) can be and regularly is reproduced by modern people. But somehow you never see modern people jogging down deer and killing them - even with the benefits of modern footwear, portable water containers, a carbohydrate-rich diet for energy, and GPS trackers.

somehow made it into popular science

The "somehow" as far as I can tell is the David Attenborough documentary bit that supposedly shows a Khoi-San hunter doing it. Richard Lee and a team of Harvard anthropologists extensively studied the !Kung (a Khoi-San people) during the '60s and '70s and there was never a mention in any of the literature this produced about these people engaging in persistence hunting. What they did describe was the practice of hunting with poisoned spears and arrows and then tracking the wounded, poisoned animal for days until it dropped and could be butchered. Needless to say, this is not persistence hunting.

The popular anthropologist Marvin Harris also featured Krantz' work is his final book Our Kind (which is where I first heard of it), but I don't think enough people read that book for it to have been the source of the idea's current popularity.