Adam, I think your argument is flawed.  Once the last Neanderthal died about 1,600 generations ago, no new Neanderthal DNA could enter the homo sapiens population, but by random reassortment, bits of it would be deleted each generation.  Sometimes, both a mother and a father would pass none of their Neanderthal DNA to their children, and then that bit is gone.  Unless, that is, there was some strong survival advantage for the lucky children who got it.  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.   Neanderthal DNA remaining in the human population should get diluted with each successive generation.  Why it survives at about 1 and 1/2 percent is still a mystery.  Many clades of archaic homo sapiens DNA went extinct and can not be found in living homo sapiens.

If (hypothetically) the last “archaic Ashkenazi Jew” went extinct 1,600 generations ago, you would not expect any archaic Ashkenazi DNA to be found in the remaining population of homo sapiens, because they would not have been a large fraction of the total human population at that time and their DNA would be diluted over subsequent time.    That is, unless archaic Ashkenazi DNA conferred a strong survival advantage.
So, are the DNA anomalies that started this thread explainable only by endogamy or something else?
Jeffrey Herrmann
New Rochelle, NY