"If I switch to a synth fibre, I notice that the shirt works as
designed (it wicks my sweat off my skin)
but so what? At that point, what (if anything) is going to cool me
down? The sweat is now evaporating off
my shirt, and not off my skin. I might end up with a cool shirt, but my
body underneath is still producing
sweat in order to cool itself off. " - racer

You have been getting alot of opinions (which are correct) but I think
this is the answer you want.
It is all about thermal transfer. Why does evaporating sweat make you
cooler? Because the heat leaves your body with the evaporating sweat.
Therefore, wicking the sweat through a shirt performs the same action
as sweat evaporating. The water, which retains your body heat is
removed from your skin (cooling you down), and then evaporates into the
open air.

Cotton holds the sweat (and heat) against your body, while the sweat on
the outside of the shirt evaporates, leaving hot sweat still in contact
with your skin. The fact that you don't dry off shows that the water
in mass quantities is evaporating much slower and retaining body heat
longer.

"Cotton is sometimes described as "dangerous" and "lethal" to wear in
survival situations, because of the hypothermic effect it has on the
body. But isn't this EXACTLY what I want when my body's overheating?" -
racer

No, because of water's ability to hold (or resist) thermal energy. In
this situation, where you are covered in damp cotton and freezing, your
body will cool to the temperature of the water. Skin does not resist
thermal change like water does (your body temp lowers to equal water
temp. instead of heating water to equal body temp).
Inversely, the hot sweat held against your skin (heated by the sun
also) will raise your body temp to equal water temp (unless evaporated,
see above).
In the