I believe that 1 in 200 people in the world today is descended from old Genghis.
The problem is that the Human genome is very large, Human Beings are a great deal more similar than they are different and Human generation sizes are quite long. I know it seems to make logical sense - if the healthiest males are killed before they have children, their genetic heritage is not passed on, and it is up to weaklings and deviants to breed the next generation. Aside from the fact that beliefs and pyschology and mental prowess are only partly genetic in origin, these things take an enormous time to affect.
For example. If all the people who have multiple sclerosis were forbidden to have children, then yes, you would indeed halve the incidence of multiple sclerosis - in 10,000 years! (By which time there will probably be a cure).
Also, natural selection is utterly ruthless. If you have a deleterious gene that is going to get you killed, it is immaterial that the rest of your genes are totally wonderful. It therefore follows that by keeping people alive who would otherwise have died young, we may very well be increasing the numbers of bad genes, but we might also be increasing the number of good genes too. The classic example would be Stephen Hawkings. In “normal” circumstances, he would have died long before making some of the greatest advances in physics.
This is the crux of the scientific argument against eugenics. The genetic composition of any individual is derived from that of their parents, but in an utterly unique way, because of the way DNA is recombinated and because of random factors (mutations). In determining what your kids are like, you are pretty much throwing genetic dice. In other words, just because you are tall, intelligent, healthy, courageous and moral provides no guarantee at all that your children will emulate you, or even much more of a probability that they will. In any case, environmental conditions (nurturing etc) dwarf genetic considerations. Especially in modern industrial societies.
To suggest that the Scots (or anyone else for that matter) are degenerate because the most adventurous/desperate of them departed a few generations ago, to colonise or fight the world, is over simplistic to say the least, and frankly racist at worse.
Holy cow: it's the race card in a scientific discussion.....
We'll never know the truth. Any conclusion is hostage to the racial politics....