Very skeptical of this hypothesis. The evidence suggests early humans subsisted on plants far more than animal meat. Likely what meat was procured did not entail chasing gazelles or antelopes for miles on end, but rather hunting for rodents and such. Or setting up traps.

Second , if this were true, we would expect all or the vast majority humans to be good at endurance running, but there is considerable variance of individual running ability. In high school, for example, as I can recall from personal experience only 25 percent of the boys in class could run a mile at a decent time and only 25% could not complete it. Those who failed were not necessary obese or otherwise had obvious medical conditions like asthma, but simply lacked the stamina for whatever reason to finish it without stopping multiple times.

Endurance running is more of a consequence of bipedalism and increased height and lighter bone mass, the ability of which is highly unequally distributed, not that enduring running arose from evolutionary pressure. As humans became lighter and less stocky, endurance running became possible.