Keeping your head cool implies that the top of the head, the part with the hair on, is where the heat loss happens.

It isn't, as you said. So dunking your hair won't make too much difference, especially since you correctly pointed out that the non hairy bits are where heat loss is concentrated.

The nose and ears only have that property by virtue of sticking off the actual head anyway.

Keep your hands and feet cool, and your face, but only soak your hair if you're into wet clothes (which is not good for your skin and can cause soreness) and ruined fabric furniture.