What I've read so far about the Anglo-Saxon migrations into England is that they were a warrior culture who became the dominant ethnic class in England, whose language completely supplanted the native Brittonic language prior to the Anglo-Saxon arrival, forming the nucleus of what would become the English language. But is this 100% the case?
As I'm more acquainted with Mainland Southeast Asian history, I'll use the example of the Thai language, which has been massively influenced by Cambodia and the Khmer language. Past historiography of the region was completely devoted to warfare and bloody conquests by various polities in the region (Ayutthaya conquering Sukhothai and Angkor, Burmese Pegu conquering Siamese Lan Na), raiding each other's people, bringing their cultures and traditions back to their kingdoms. Recent scholarship, however, is beginning to point out that the transfer of cultures and languages between the various peoples in the region was as much friendly and consensual, through trade and diplomatic marriages, as it was through violence and warfare. What I find is that there is almost never an instance where a culture has been 100% supplanted by an invading culture in an instant or clean-cut way.
My point is that it's been implied for a long time that the Anglo-Saxons completely and instantly wiped out the ruling class of Post-Roman Britain and supplanted the native culture and language with their own, through warfare and subjugation, which doesn't help things as the past "nation-state" historiography of Britain, and in popular/mainstream history up to this day, glorifies the Anglo-Saxons as the founders of English culture, in essence. So what is there remaining of pre-Anglo-Saxon culture and language in Britain today (or what had initially survived the Anglo-Saxon migrations for longer before altogether disappearing)?
You might be interested in the article 'Why don't the English speak Welsh?' by Hildegard Tristram. You can find this for free online by searching the title. She answers a lot of the questions you've asked.
But yes, the Anglo-Saxon invasion is often seen as an elite replacement today, which did have fairly long lasting effects on the landscape of Britain. The extent of this is fairly unknown.
I'd suggest my answer here about DNA in Britain which explains some of the overviews about the Adventus Saxonum and to what extent the Anglo-Saxon invasion was a rapid cultural change brought upon Britain: https://www.reddit.com/r/AskHistorians/comments/tge62i/-/i11uo9d
In summary, there's two theories that float around, one of continuity and one of discontinuity. Discontuity states that the arrival of the Saxons sparked widescale and rapid cultural change due to apartheid, subjugation and death of the British populace. Continuity theory states that the Britons were largely unaffected by the arrival of the Saxons, with many being initially left alone or the Saxon leadership being so similar to past rulers, and there was a period of slow, consistent ethnic and linguistical change. A third option is a mixed based approach, which states that any/all of these may have happened to various degrees in Britain.
Now Tristram, who I mentioned earlier, goes into a lot of detail about Middle English, which is the English language that starts appearing after the Norman invasion. Now through investigation, she finds that Middle English is most probably the bastardised form of Old English that the native Britonnic non elite learned. They probably tried to learn Old English to fit in politically, or for other various reasons, and thus added their own various flairs to the Old English language. Though these flairs wouldn't be seen until the Norman conquest, because the Saxon elite had been incredibly good at keeping writing for only those at the very top, who'd had an elite education, and were taught a standardised strict form of Old English. One which may have been very different from the language spoken by most of the populace. Thus when the Saxon elite is disbanded by the Norman conquest, we start to see this bastardised form spring out into daylight via Middle English, which would then slowly morph into modern English.
So, yes, there is a lot of Britonnic influence in modern English. Not necessarily in place names, as Tristram notes, but via syntax and grammar.
This is in line with modern scholarship that is starting to move away from 'Saxons came and absolutely destroyed Celtic culture and bombed it into the abyss', but rather that the Saxon invasion saw a cross fertilisation of both North Germanic and Celtic culture. My answer above mentions some of the Saxon kingdoms such as Mercia and Wessex having possible Celtic roots (especially Mercia), which may have 'Saxonised' through far more peaceful avenues such as Royal marriage, alliances and trade.
So in conclusion, the Adventus Saxonum is very difficult to account for, considering (as mentioned in my other answer) we cannot distinguish it from earlier (Neolithic) and later (Viking) Germanic invasions, and Roman material culture was quickly replaced with Germanic material culture in Northern Europe following Western Rome's collapse. So we have a very murky picture of what happened in Britain, because the Celts are leaving behind almost no trace of their existence. They're materially invisible due to Germanic material culture being so dominant, so any burial sites will show them buried with Germanic goods, and in terms of literature they're not being published, so they're 'culturally invisible'. They basically left us almost no sources, and thus we had to rely on very shady sources like Bede who shows us more what the Saxons wanted to see their past as, rather than how it actually happened. Especially considering Bede is writing an ecclesiastical history, so actual history isn't really his main goal. He's trying to write a religious piece. Thus, the Britons' effect is only starting to be seen with scholars like Tristram finding Celtic influences in the syntax and grammar of Middle and Modern English.
This isn't to say that the Adventus Saxonum wasn't a massive event. It probably was. But it wasn't an absolute destruction of a people's culture and language as previously thought, and we have to try and assess the extent of Germanic goods and culture seeping into Britain regardless of the event, to then start to assess how major the event actually was.
My sources are linked in my other answer if you're interested in reading more.