Given that this is a) perhaps the most important single question in early English history and b) orthodoxy has been working from a false premise (that the English language is a modern form of the Anglo-Saxon language) it is not surprising that Orthodoxy tends to flail around a bit.
One school (the “exterminationists”) argue that since language usually only changes when the population changes, it follows that the Anglo-Saxons killed most of the Ancient Brits and drove the rest into Wales where they became the Welsh.
The other school (the “integrationists”) argue that no, under certain circumstances language change can happen naturally, so they put the emphasis on lots of intermarriage with the kids being brought up speaking Anglo-Saxon plus people learning Anglo-Saxon “to get on” plus Anglo-Saxons breeding faster plus, well, anything they can think of which tends to have the desired effect because…..
THERE IS NO EVIDENCE. You can say anything you like without the slightest fear of contradiction because we don’t have any physical evidence—historical or archaeological—that bears on the question. (Yes, this applies to me too, but as my critics are fond of pointing out, I don’t use evidence….)
On the whole, which side is prevalent at any time depends on the general political climate. Thus the integrationists are in the ascendancy in these liberal times but of course everybody not directly involved in the controversy uses the liberal version of Hegelianism: “Oh, I’m sure it was a bit of both, darling, one mustn’t be dogmatic in these matters.”
But one can be just plain wrong….darling.