It's not my premise, but Colin Fletcher and Chip Rawlins' premise, from the Complete Walker IV, if I recall. And it makes perfect sense. The Tilley LT I have is not waterproof - it's synthetic and breathable. Natural fibers are breathable as well. I use the synthetic Tilley for the weight savings, water resistance and quick dry properties. A good head covering or hat will keep a pocket of air around your head. And the total shade is important - the sun's energy will really cook your egg. The fact that a pocket of cooler air is kept near you is what matters. How would this change with synthetic or natural fibers, if both are breathable enough to let moisture out?
BTW, I have lived in the desert for weeks at a time, carrying a lot of gear, wearing a similar style boonie hat that keeps a pocket of air above your head but still breathes, and have spent time with desert nomads, and they have learned a thing or two after living for so many generations in such harsh conditions. They wear wool and cotton (and sometimes even an insulated synthetic jacket in the cold desert nights).