Amount of melanin decides the darkness and fairness of skin. The darker the skin is, larger the heat energy it will absorb from sunlight. The lighter the skin is, less will be the heat energy absorbed. The people in cold regions of world should have darker skin to absorb much heat available. And people in hot climate should have reflexive white skin to reflect much heat from sunlight. But here's exactly opposite of it. Why?
Your conclusion is actually the exact opposite of the reality.
Light skin in colder climates (i.e. climates that have short winter days) is selected for because melanin blocks the absorption of 310nm UV light which is needed to produce vitamin D.
The evolutionary significance of vitamin D, skin pigment, and ultraviolet light; RM Neer,
Am J Phys Anthropol. 1975 Nov;43(3):409-16.
The less melanin you have, the more UV you absorb, the more Vitamin D you produce. As Neer says, "Vitamin D deficiency can cause death, immobilization, or pelvic deformities which prevent normal childbirth." Vitamin D deficiency can be fatal, so the harder it is to produce, the harder it is to produce Vitamin D, the less likely you will be to survive and reproduce in areas where days shorten for 6 months out of the year as a result of the tilt of the earth.
This can be witnessed in the lengths that societies living closer to the poles go to to supplement natural Vitamin D production, adding Vitamin D to dairy products or as this famous National Geographic photo shows , exposing children to doses of UV light.
- Ultraviolet Bath, Russia by Joe McNally
This article, The evolution of human skin coloration by Jablonski and Chaplin gives a review of the how skin color is an adaptation "related to the regulation of ultraviolet (UV) radiation penetration in the integument and its direct and indirect effects on fitness."
Your underlying assertion of the purpose of melanin is incorrect. Although we need sun exposure for production of things like vitamin D, it doesn't take much to do this. You're correct that lighter materials will reflect more light, but that's not enough to prevent UV damage. So there's no advantage for being able to absorb more sunlight for energy or protein production.
Instead look at it as a protective mechanism. People with light skin suffer from UV burns in shorter time than those with darker skin. People in cooler climates are typically in the sun less, or at least more of their body is covered when they are in the sun. They don't need as much protection from the sun as do people where it is warmer that are typically out more with less clothing needed to feel comfortable.
Many times the claims that "disprove natural selection" are just because they're being viewed from the wrong vantage point.