It would surprise me if one common ancestor only 6,000 years ago could be responsible for giving a half billion people a recessive gene. The previous theory of Neanderthals having fair skin and blue eyes for 500,000 years and then integrating with homo sapiens seems far likelier.
My late husband had intensely light blue eyes. He was red haired, pink skinned, big boned and very muscular. A weak chin and heavy brow, and short intense temper. After all this Neanderthal stuff came out I concluded he had at least 4% if not more Neanderthal blood. Genetically he was Scotch/Viking. About 40% of northern British have the “ginger gene”. I have hazel eyes with some green, some brown, and some dark blue. Both of our sons have deeply brown eyes. I need to check on the eye color of 4 grandchildren and the wives. I had a semester of genetics in college. Many do not know that there is a progression in the strength of gene alleles. I think that light blue is the most recessive, Asian brown next most recessive, and other colors less recessive but I don’t have a guess as to in what order. I have some Asiatic tartar blood from my Prussian German petit nobility ancestry on my mother’s side. What I think allowed the dark brown of my sons’ eyes is that they each got the recessive brown Asian gene from me (with the European genes providing my hazel color), and the most recessive pale blue eye gene from my husband which was submerged by the stronger Asian brown gene.
Regarding the rapid spread of the light eye gene, it could be from Neanderthals or a later source. At any rate blue eyes usually go with fair skin and fair skin especially in the north allows more formation of Vitamin D on the skin and absorption to grow strong bones and wide hip structure conducive to easy delivery of live children. It probably did not take men too long to realize that fair, wide hipped ladies had a higher rate of successful births.