The only caveat to this is that you then need to fast forward 1,000 years & as someone so removed from present day decide how you are going to categorize the study of North America in the 20th & 21st centuries.

These divisions are not made by people at the time, it's all done arbitrarily by historians hundreds of years later in order to aid the studying of the subjects.

There is an argument for the separation from the Roman Empire but it's not the points you are making. You are personifying an imperial state & statements like they didn't have respect for the latin empire is just plain wrong.

People who make the case it should be referred as the Roman Empire often give the argument that people themselves believed there was continuity & identified as Roman. What they forget is that historical naming conventions never take that into account, they are arbitrary dividing lines used to draw up history into manageable chunks & bring attention to significant monents of change, for example the decline of the Roman Republic. It's not a reference to people's identity.

However there is also a valid case to make that Byzantine Empire is a pejorative term that has too much negative baggage & should be discounted. The term itself has become an insult to refer to courts that are back stabbing, conspiracies & low morality. The consensus among historians is to refer to it as the Eastern Roman Empire & to do so at an earlier date, usually from Theodosius, to emphasise the continuity whilst still making the historical definition. Popular culture such as video games have not caught up with this however.

The exact definition of where you draw the line is difficult to identify clearly. It's not like the end of the Republic where we can draw a line when hereditary rule starts. Changes occur gradually by different emperors over hundreds of years. But there's no real need to be exact & articles with opinions on this are entertaining reads that anyone interested in the period would happily read so it's not exactly a problem. It's helped that Ancient Roman historians often get off the bus when he hit Constantine & the empire becomes Medieval in structure.

What we should really do, in my opinion, is make more divisions. Carve up the Roman Empire into several empires of different rise & falls. Harriet Flower has an excellent book that does this for the Republic arguing that Rome from a historian studying perspective had not a single republic but 6 distinct republics with definably different structures of govt, visible rises & falls and also experienced two interregnums where govt broke down.

Accepting that these lines are just divisions by historians & with that chopping individual periods up further would do alot to boost the study of those periods. For example historians look at Augustus's state and ask the question, how did the Roman Empire fall, they then start to talk about the Goths & migrations, disease, taxation still in the context of the Augustan Empire. This inevitably leads to historians giving all the change from Augustus to Honorius as a cause of the fall. This isn't correct, the question isn't framed right and should be always in the context of the Empire at the time which was very successful despite its differences. If we draw our lines here and say the Principate/Augustan Empire was its own distinct historical entity with a rise and fall that becomes easier then the smaller scope makes it easier to define. The Augustus Roman Empire fell during the crisis of the Third Century when the succession mechanism broke down. This line also gives us a nice valley for additional debate to flood. Maybe some revisionist historian wants to point out actually they can evidence it just took so many generations for inflation & economic damage to hit & succession wasn't a big part. Etc.

This is why we have these lines as historians & in my opinion the more the better.

Whilst you are right that initially Britania continued as under the Romans it did not remain that way up until the formation of the Anglo-Saxon kingdoms. We know large provincial authorities fracture at some point. It is unusual for this to happen peacefully & having independent regions the size of towns almost always leads to struggles with raids, local resource conflicts etc.

We know the wall stops becoming defense & Celtic raids become more frequent with places like Vindolanda - which carried on as a civilian bathhouse after the troops left, was abandoned. We also can see the coin deposits found that were buried & never returned to.

Whilst it's true large towns & cities, the best places for finding evidence of violence in archeological record would have been largely insulated from this. It doesn't mean the countryside, where wooden buildings leave little record isn't being impacted.

So yes, Britain carries on as before when the military forces leave & do not return but we see a slow gradual decline of authority to local struggles towards the end of the next century.

I haven't read either of the authors you mention but I know those that argue against any form of migration often use straw man arguments by using a mass invasion as the opposing argument they are refuting. They also heavily refer to Bede's explanation & argue against that. The problem with this approach is we know both the mass invasion hypothesis & Bede's description are untrue, refuting them doesn't refute migration.

Here are a few things we know:

1. Bede lived in a world of defined Anglo-Saxon kingdoms, he tried to explain how they came to be using mass migrations so his theory of Jutes south east, Saxons in the South, Angles in the mid & North is him working backwards rather than actual historical research. This is both proveably false & totally implausible that it's easy to take shots at. Just because migrations definitely didn't happen how Bede describes doesn't mean they didn't happen.

2. We know the late Roman Empire used Germanic tribes to supplement it's military forces & that towards the very end significant military towns in the Empire were drawing their troop requirements repeatedly from specific Germanic areas. It is believed Britania was doing the same with the region that contained Saxon tribes & this would likely have continued when Britain was cut off from the empire. As the military latin used by troops in Gaul created French the military Germanic language used by Saxon tribes could have had the same effect in England. The question here would be why the tribes would keep Germanic language than learning Latin & the answer is probably explained by the fact Britain was cut off from the Empire & so Latin wasn't as significant for all soldiers to learn.

3. Though Bede's explanation of how can be debunked what we cannot ignore is that Bede of many cultural touch points that refer to a migration of some sort. Of Germanic ppl's coming over to Britain. There is something in living memory of that experience both in the Anglo-Saxon culture & in the Welsh & Cumbrian cultures.

4. We know through studies of the fall of the Roman Empire & of colonialism that languages & cultural identity don't need a population displacement to change root & branch. In a society where military might equals total power & those with it were the minority it actually wouldn't take a large number of people to migrate to see a cultural change if these dominated the military class. The more dangerous & fractious society the more this is the case. So if it's a period of lack of central authority, there are small petty kingdoms everywhere, villages are fighting villages. Then a member of a Saxon warrior band sets up farm in your village. You can bet your bottom dollar every villager is kissing his ass & learning his language to both avoid being his victim & to use his warrior influence to ensure their farms are also defended. Just like that he's the feudal lord there. Individual ppl are chameleons who will copy others culturally if it improves survival or offers advancement.

5. The best Roman coin hoards we have found in Britain come from this period. Though it's not uncommon for coins to be buried for safety in normal times it isn't common for large coin hoards to then be forgotten about. These are instances where a person has buried a hoard of coins & then never been able to go back & claim them. This is signs of fleeing suddenly.

As a result of these facts the historical consensus at present was that there were small warrior bands & traders migrating & settling in England. These were usual of mixed tribal background, warrior bands tended not to be ethnocentric. These bands were military elite & drove immitation. It was a violent time of which these bands were part of that drove large scale uprooting of population - note not necessarily driven away by invading migration but by the violence of the time of which the warrior bands took part. Though some must have been displaced by settling bands. What emerges from the violence is a society that has homogenised around the Saxon warrior elite.