Nope - this isn't how population genetics and natural selection work in real life on the timescales we're talking about.
- I oversimplified for the sake of having an illustrative example in my previous post and threw everything under the banner of "SNPs", but most of the DNA we inherited from Neanderthal populations is non-coding. We have huge amounts of non-coding DNA in our genomes, and this DNA often persists in our genomes pretty much forever unless there is some unusually compelling selective pressure interacting with whatever it does (and, being non-coding, it often doesn't do much, and what it does do is often pretty subtle). Included among our non-coding regions are transposable elements that have been littering our genomes for millions of years. Our genomes do not do rapid evolutionary cleanup on these regions; they just...stick around, for the most part, as long as they don't end up causing some sort of deleterious mutation. My larger point here is that human evolution is much less tidy than the model your understanding appears to be based on.
Unless, that is, there was some strong survival advantage for the lucky children who got it.
- If the principle "new DNA always rapidly disappears unless it confers significantly increased fitness" were really inexorably true, then we would never see recessive-gene-linked diseases persist in populations - these mutations would rapidly disappear. In reality, they stick around for quite a long time in populations, and so do the parts of our genomes that stem from Neanderthals.
- The science on what, exactly, is in Neanderthal DNA is still in its infancy, but many researchers are increasingly convinced that at least some Neanderthal DNA did indeed confer valuable adaptations on populations that ended up with it (for instance, on immune system function).
- Exactly what happened over time to the Neanderthal-derived parts of our genomes is also a subject of much recent debate among scientists. But there is at least one empirical study in the last five years that suggests that the initial introgression of Neanderthal DNA faced a pretty rapid initial purge, but then largely stabilized. (The article reviewing this study doesn't do a deep dive on why this would have happened this way, but the model makes at least some intuitive sense to me: maybe the initial selective purge was of various genes that coded for traits that were obviously deleterious for Homo sapiens sapiens for one reason or another. After those disappeared, there was still a significant amount left that was either beneficial or innocuous, and consequently those bits of the Neanderthal genome faced little to no selective pressure. So those mostly stuck around in the population.)
And there is no reason to believe that when the last Neanderthal went extinct the number of homo sapiens/Neanderthal hybrids was large compared to the number of pure homo sapiens.
Far from obviously true when limited to the populations we are talking about: the relative number of Neanderthals vis-a-vis the initial populations of early European
modern humans who they interacted with.
Estimating the size of Neanderthal and EEMH populations appears to be a tricky business. But some quick checking suggests that the total number of individuals in each might have been roughly comparable, at least at some times. I found one figure suggesting ~3000 Neanderthals about ~55000 years ago and another listing an average (with a wide upper/lower bound) of 4400 EEMH around 40000 to 30000 years ago.
That suggests to me that it's not absurd on its face to think that at the time EEMH and Neanderthal populations were intermingling, the relative size of each population could well have been similar enough, and small enough, for Neanderthal DNA to plausibly enter into, and then spread throughout, the EEMH population within a few dozen generations of each interbreeding event. Exponential growth means that there wouldn't have had to be all that many interbreeding events for this to happen!
Many clades of archaic homo sapiens DNA went extinct and can not be found in living homo sapiens.
That may well be. It's not impossible for there to be evolutionary dead ends - a population that diverged from our ancestors, became isolated to some degree and formed its own genetically distinct branch, and then died out. In this case, two populations that diverged genetically from a common ancestor (probably H. Heidelbergensis)
intermingled again, their descendants survived to eventually develop agriculture, bronze tools, third-wave ska, and the pet rock, and we retain the DNA of both to varying degrees. And?