If losing most body hair helped humans cool down, why did hair remain on the head, eyebrows and eyelashes?

Answer: Those patches still solve local problems

Those patches still solve local problems ✓ — Correct! Hair loss was not a plan to make every patch of skin bare. Scalp hair shades the head from strong sun, while eyebrows and eyelashes steer sweat and dust away from the eyes. Evolution can reduce hair where it hurts cooling and keep it where the local benefit is still worth the cost.

Evolution had not had enough time — Wrong. Humans have been mostly bare-skinned for well over a million years, long enough for selection to remove the remaining hair if it only got in the way. It stayed because it still does a job where it is.

Any remaining hair is just genetic noise — Wrong. Scalp hair, eyebrows and eyelashes are not leftovers. Each sits where it solves a local problem, shade on the head, sweat and dust kept out of the eyes, so selection kept them while reducing hair elsewhere.