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."