No.

Nope. Your brother is wrong and his 'historian' friend may be an idiot. Let's address his 'narrative' before we look at evidence. Even today, around 30% of the population can claim Germanic heritage. On the East Coast and in the North this rises to around 40%. When you factor in that around 10% of the modern population is BaME, mostly since 1945, that proportion grows even larger. This tallies essentially with the projected levels of Anglo-Saxon colonisation in England, allowing for a 10% increase in regions where not only was there Scandinavian (and thus also Germanic), but also higher levels of Anglo-Saxon incursion.

When the Vikings arrive in 793, the Anglo-Saxons have been established for well over 200 years. While the Danelaw was established over much of North-Eastern England from the 880s til the 910s, this was not a period of ethnic cleansing. Alfred of Wessex in 886 makes reference to liberating 'the English living under the Danes'; had an ethnic cleansing occurred, we can be sure that contemporary English sources would have referenced them to highlight 'heathen' barbarity. The Danes in England rather largely assimilated into their host society, the evidence for which we'll come to later.

There are two instances of what we might call ethnic cleansing and one is carried out by the English on the orders of Æthered II in 1002. The St Brice's Day Massacre, as it's known, was an attempt to purge the ethnically Scandinavian population from much of Eastern England, and went down badly overseas. It illustrates, however, that the Danes had assimilated into and not replaced the local populations in the 880s. The second instance, what we could loosely term ethnic cleansing, is the 'Harrowing of the North' carried out by William the Conqueror in the 1070s. A terror campaign designed to quell rebellious Northumbria, the harrowing occurs precisely because the Normans form a relatively tiny elite, and their hold on power in the decades immediately following the Conquest of 1066 is often tenuous at best. While the Normans did by-and-large replace the Anglo-Saxon nobility, they had nowhere near the numbers to even threaten to replace the population. Often, the Normans, like the Danes before them, assimilated into English culture; They married the widows and daughters of the English and Welsh nobility. Domesday Book, compiled in the 1080s shows us that while many of the new lords of manors were Norman, a fair proportion of Anglo-Saxon names remain, especially in the Church.

we have only a few ambiguous scraps of evidence

This is categorically wrong. While it's not huge compared to, say, the 15th Century, we have a significant corpus of evidence from the Anglo-Saxons. A vast amount of this is archaeological. Your brother and his idiot friend may like to visit the British Museum or go see the Staffordshire Hoard in Birmingham, just two examples of large collections of Anglo-Saxon material. The numismatic collection of Anglo-Saxon coinage is extremely huge. Anglo-Saxon society was fairly extensively monetised and as a result, a large amount survives. The Syllogue of Coins of the British Isles and the Fitzwilliam database of Early Medieval Coins contain enormous collections of Anglo-Saxon pennies. The Portable Antiquities Scheme is another hugely valuable resource which contains thousands of not only coins, but also myriad examples of tools, jewellery, clasps, horse tackle, personal wear and general metalwork used and lost by the English. We have whole corpuses of metalwork, various productive sites, fashion trends and their continuity. We know, for example, that the West Saxon fashion for Cornish/Brythonic inspired 'Trewhiddle' pattern designed strap-ends spread across England in the 10th Century, and in turn inspired the production of Danish 'Borre' designs which then spread to Scandinavia but also became fashionable in the Midlands. We know that Mercia eschewed the use of ceramics at the same time as new styles were springing up across the Eastern Midlands in the 900s, until Æthelflæd mandated a Roman-inspired series of ware types that were used to distribute military rations from the major productive site at Stafford.

Our literary sources from the period are also numerous and far more than 'a few ambiguous scraps' and Beowulf. Bede's Historia Ecclesiastica, The Anglo-Saxon Chronicle, the Mercian Register and Asser's Vita Ælfredi, to name a brief selection, are all Anglo-Saxon histories. The Anglo-Saxons appear in contemporary Welsh history too, in the Annales Cambriae. Later historians such as Florence of Worcester and William of Malmesbury cite now-missing Anglo-Saxon histories in their works, which suggests that there was a much wider corpus, and also shows us that the Normans had no interest in erasing the Saxons.

We have surviving legal documents, including legal codes of Æthelberht of Kent, Ine of Wessex and Alfred of Wessex to name just three, we have a significant corpus of wills also, and a vast library of Charters accessible via Sawyer's Anglo-Saxon Charters, many of which are now digitised. We have administrative documents (The Burghal Hidage), we have phrasebooks (Ælfric's Colloquy), we have medical textbooks (Bald's Leechbook) and collections of riddles and poetry (The Exeter Book). The very fact that we speak English as a language, despite all of its French influences, is overwhelming evidence for the.continued survival of the Anglo-Saxons.