Did you read Dennis Venema’s series on this topic? I think his explanations are very accessible. This is the first part:
I’m not completely on top of the matter because I’m not trained specifically in genetics, but I’ll try to explain it according to my understanding derived from readings of Dennis Venema. I do have experience with genetic algorithms though, but that’s a different part of the story.
I think the argument is mainly derived from information theory. Today, there is a certain amount of genetic diversity within mankind. Based on that, you can place constraints on the minimum original size of the population, using the rate of reproduction. You even don’t need to make assumptions about the mutation rates for that. Dennis explained (in this post) that the genetic variation in human populations is far too large for having resulted from a single ancestral pair in the past 200,000 years or so.
The assumption of ideal mixing is actually very conservative. Assuming restraints on the mixing (as in nomadic populations) would result in even less variation and therefore it would push back the limit even further in time. This would make the likelihood of a single ancestral pair even smaller.