Is my theory of probability and ancestors valid?

I am an anthropology student from Michigan. I am researching the inter-relatedness of most if not all humans as part of my studies. I personally have an interest in statistics and probability theory and I have used it in regards to this topic. Before I explain my theory, I received the following from a computer scientist that I wish to share to better give a background idea of what I’m trying to put forward:
If there were random intermixing, then we would each have ~1 million ancestors living in 1500 AD, out of a world population of ~500 million.
So the fractional overlap between two people would be about 1/500th.

But the probability that two people share at least one common ancestor would be essentially 100%. Basically, you are choosing a random number between 1 and 500 a million times and you're asking whether you ever choose number 500. In a million trials, we expect this to happen 2000 times. So that it happens at least once is guaranteed.

If we get rid of the random intermixing, the fractional overlap will drop to much less than 1/500th. But I suspect that the probability of at least one overlap will remain very high.
If the population in 1500 was 500 million, and it is 6 billion today (12x larger).
If the average generation length is 30 years, there are 17 generations in 500 years.
So the average number of surviving children per mother is exp((log 12)/17) = 1.157
Since a child has two parent, the average number of surviving children per person is 2 * 1.157 = 2.315
So this is the average growth rate per generation for the descendants of a person in 1500.
2.315^17 = 1.575 million. So an average person in 1500 has about 1.5 million offspring alive today. Sampling from the whole world, the probability that a random person from 1500 is an ancestor of a random person in 2000 would be 1.5 million / 6 billion = 0.025%.
If you were only considering people in a region like Europe, it would probably be something like 1.4 million / 700 million = 0.2%.

As I have said above, this is from a computer scientist. Next, is my own theory. From a probabilistic standpoint, due to the Law of Large Numbers, me, you or *almost anyone alive today had an ancestor in like say China or almost anywhere in Asia in 1500 CE. I chose the year 1500 because of each person had in theory about a million ancestors- lower than that because of inbreeding but still a sizable number nonetheless. I have shared this theory with several of my colleagues and they are intrigued by it yet I have not yet shared it with an actual statistician. I know it seems unlikely, but from a strictly statistical standpoint, what do you think of it?

*please note, this excludes populations that have not have had contact.