Correct, its a big benefit over the furry prey. Especially when chasing the animal in the hottest part of the day for hours, pre-modern people had quite a different life.
You know I've thought about this, and if you look at it a certain way, I think you can say life is no different.
We still wake up as a group. We still "hunt" except the payoff is money convertible to food. We still eat together. We still sleep together (just across from one another in houses).
It's neat to think about.
But, I don't think it matters much. It doesn't help, but it's a small surface area.
It's certainly a hypothesis I've long held but never had chance to test.
Citations for support/contradiction welcome, thanks.