So trying to prove that the population changed in Britain because of the Anglo-Saxon invasion using genetics doesn't work?
Yes.
Now, we
know that some people migrated to Britain from the Continent in the fifth century, and that many of these people probably spoke a Germanic language, which eventually became Old English. So much is clear. For lack of a better term, we usually refer to them as the Angles, Saxons, and Jutes. But we do not know in what numbers they came, or why; nor do we know what they did when they got to Britain. We do not know what happened to the speakers of Latin and Brythonic in the English lowlands, who numbered in the (very low) millions at the beginning of the fourth century. There is an enormous gap in the historical narrative for Britain from 410 to the early seventh century. By 600, there were several large polities in control of most of what is now England, which are usually classed together as "Anglo-Saxon" kingdoms. Explaining how they got there is the tough part.
Now, we know that the population of late Roman Britain was, for lack of a better term, too big to kill. The Angles, Saxons, and Jutes cannot have numbered even into the tens of thousands, due to the agricultural limitations of their supposed homeland; it would be essentially impossible for them to have massacred the Romano-British wholesale. It would be an instance of genocide on a scale unparalleled until literally the Holocaust - but it took a great deal more than a few thousand Nazis to slaughter the Jews
et al. We can safely dismiss the idea of genocide and resettlement.
All the same, the fact remains that, two centuries after the revolt against Constantinus "III", there were several kingdoms well entrenched in Britain dominated by people that spoke a language unknown in that island during Roman times. So some sort of migration
had to have happened. But the method by which these Angles, Saxons, and Jutes seized control of most of the island is completely unknown to us.