Why do you need such hair to avoid overheating?...
Hairless primates, namely humans, that have cooked food, and thus do not need huge gut with long transit times to digest food eaten raw, as apes do. So apes have relatively small surface to volume ratio (compared to "hairless apes") for cooling. Thus, humans can more efficiently dump more heat to the environment, especially as even weak air currents, unimpeded by a cm of thick hair (or fur) increase the rate of cooling by evaporation (perspiration).
Digestion of food eaten raw, is much more energy intensive process than digestion of cooked foods, which have been soften and partially broken down into smaller molecules by the external heat (stove or fire). Thus not only is there many more cubic centimeter of "ape gut" than human gut, the energy each cm^3 uses to a break down complex molecules into smaller units that can pass thru the intestine walls, into the blood is greater too. Then there is also the fact that the ape gets a smaller fraction of the calories from the raw food with his less efficient digestion of it and it takes longer , still requiring energy before it leaves his body.
All these factors make the apes thermal problems worse as his cooling surface to gut heat production is a smaller ratio than that of human. I doubt an ape could survive in the full noon sun light, if you shaved all his black hair off and let almost all of the solar heat of the African plains reach his skin.
A very high fraction of the sun's heat is absorbed by the ape's black hair and quickly transferred to the air, not the ape's body. Most humans don't need to block sunlight from reaching their body FOR THERMAL REASONS*; however, humans who live in or transit hot solar deserts keep most of their body covered with more than one layer of cloth. For one who does not understand what I have been explaining about the importance of NOT letting the solar heat get to the body, this use of thick layers of cloth by the Bedouin and other desert Arabs seems strange.
* Skin cancer is a different reason.
As one of my six references in post 1951 explicitly said (and all imply): Raw food and its required large volume and large energy requirement gut, leaves too little energy left over to have a big high energy demand brain too. That is also the answer to your question below, so I only summarize after it.
Huh? What requirement can not be met with raw food?
It is impossible to get from raw food the energy needed to support BOTH the raw food's required big, energy-intensive, gut and longer processing period (days, not hours, of transit time) with only a fraction of its calories released in the inefficient digestion, (done only by the body), instead of with pre-softening and some external partial break down of complex molecules by cooking; AND the very energy intensive requirement of large, human size brain. Elephants can do this as they have evolved an "auxiliary cooling system" - huge ears, and larger ears for the African elephant than the Indian elephant as African elephants are under greater peak heat stresses.
Also bigger brains (of same nearly spherical shape for shorter inconnection white mater) have a worse thermal stress than smaller brains do as both their surface to volume ratio is lower AND internal cooling by blood flow thru the brain, especially the small capillaries, is dissipative - I.e. more heat generate by the blood flow which it must remove. Frankly, I think the "thermal stress" argument as the cause of bigger brains has every thing just backwards. Please, some one, explain why bigger brains relive / reduce thermal stress.