None of these arguments are compelling. The argument in your Wiki link is that humans were too stupid to track prey out of sight, a supposition based pretty much on egocentrism. Further, all of these arguments are on the basis of one human or multiple humans running in a straight line behind one prey animal that, as soon as it finds cover, magically wins.

I'm no anthropologist. But the arguments put forth by your links so far don't take an anthropologist to question. Look - humans sweat. Sweating is much more effective under airflow. No other animals sweat the way humans do. You can point to the function that adaptation serves extremely well - movement in extreme heat - and make an argument that we evolved for movement in extreme heat. And while you can argue that maybe sweating isn't for that, without a compelling counter-argument you're just being contrary. Same with human hair. If not heat then what? Especially when you consider how much human perspiration occurs at the scalp.

The use of fire is between 1.5 and 0.5m years ago. That right there makes roots bioavailable. It also supports a radical shrinkage of our digestive tracts, a massive expansion of our cerebral function and a consequential enhancement of language and socialization. There's a lot of circumstantial evidence that points to the evolution of humans as we know ourselves as a consequence of carbs, starches and proteins due to cooking, and the influence of carbs, starches and proteins building human social behavior. Most of the arguments against these theories are not "that's disproven because" but "well isn't that convenient."

Yes. Yes it is.

Long Running need a lot of carbs. Even if you catch the prey, you get protein and some fat, but you cant replenish those carbs.

The whole of the savanna is covered in plants designed to survive fires by regrowing from their starchy roots. It could even be suggested that once you start boiling grass roots for food, you suddenly have the reserves to take up distance running.