it isn't unfair to say that both initially distinct groups both had a part to play, it is fair to say they both had influence (whilst much less precise than your terms) as such, partial.
Thing is, Barbarians were distinct more from a political point of view than cultural, even at the beggining (at least how they appeared as coherent peoples). The distinction grew out of it, especially in the late IVth and Vth century when Barbarian formed cohesive political entities within Romania, rather that the
relatively scattered settlements of
dedicatii.
You had, IMO, something dialectical at play there, with Barbarians not being really distinguishable from the provincial population they were influenced by much more they influenced them (setting the base for a stronger integration of second-wave settlements and takeovers); then a growing political distinction in the IVth/Vth centuries (while culturally, romanization continued to fully operate), and then a lesser political distinction as Barbarian took over the post-imperial institutions.
But again, the very real Barbarian influence was a romanized influence on a Roman population. The properly non-Roman influence does exist, but is really limited to fringes in most of western Romania, and took a certain time to really last.
The key point here is that whilst the differences (from our perspective) may be slight - at the time my understanding is that there were still clear in and out groups between the Romanized Barbarians and 'Romans Proper'.
Thing is, because the distinction is essentially political, you had several Romans, or long established
dedicatii that joined up with Barbarians since the IIIrd century (or, arguably, as soon as they formed coherent peoples along the limes). In terms of material culture, the difference between settled Barbarians and Romans is not slight, it is virtually inexistent.
As such, until that distinction isn't made by Roman and Romanized sources, then I don't think it is fair to not recognise the distinction that the sources seem to.
Most of distinct features appeared after Barbarians settled in Romania and weren't a remnant of non-Roman influence.
For what matter Franks, for exemple, a lot of what supposedly distinguished them from Romans are a later creations : fransciscae doesn't exist before the VIth century, the Salic Law is essentially a rebranded Roman law on Barbarians, the mythified history of Franks as written in the VIth/VIIth centuries is a mash-up of Aeneid and Exodus, the supposedly frankish tradition of shared kingship is absent in earlier centuries but essentially mirror Roman shared-imperium, "typical" Frankish clothing doesn't appear before the VIth...
To quote Bruno Dumézil, from the VIth onwards, with a provincial population more and more identifying itself as Frankish (for political, fiscal and prestige purposes), the differenciation is so impossible to do, that Gallo-Romans are playing Barbarians as some kids would be playing cowboys, fantasizing about their Barbarian identity from half-remembered stories, Tacitus and other latin writers, and inspiring themselves from populations that still existed within the Barbaricum, mixed with little romanized Barbarian features that survived the fall of the western Roman state.
And we're talking of a people that is considered having kept most (romanized) Barbarian features : it's even funnier with Goths. A slight analogy could be made (altough certainly not similar) with Hiberno-Americans rewriting their history (the stress on indentured servants, etc.), celtizing themselves in a manner that no XIXth Irishmen would have done, etc.
I'm not sure I see the distinction between break-off of an institution (at least in part) and changing those who rule on them.
Breaking-off the institutions would imply that these would at least cut themselves from each other, or from the institutional poles from which the new rulers wouldn't have control.
While we see a growing estrangement due to the collapse of the roman state, and the decline of structures that depended directly from it (altough it's far less obvious for Africa and Italy before Roman reconquest), you still had enough political and institutional continuity and exchanges between Barbarian kingdoms, Exarchates and the eastern Roman state, which wouldn't have happened if these polities really broke off,
at least as it seems to be commonly defined.
From my perspective even if the institutions operate fundamentally identically, they belong to different 'establishmetns' or heirarchies and as such seize to be the same institution as one another - one simply being previously part of the other.
It's a bit moot for what matter the Late Empire : Barbarian takeover didn't happened overnight, but was a slow take of control on key institutions (such as the military, which did included due to the times a territorial control over civil militia). With the fall of the western Roman state, it was less Barbarians taking advantage of the vaacum (altough they certainly did took advantage of the weakness of the roman state in the Vth century), than the logical consequences of the greater part Barbarians played in Roman state since the IVth (just look at the long list of Frankish-Romans generals in the west, possibly related to Merovingians).
However, I use the term 'dominant' because they took over institutions, like armies, or introduced ones loyal to their in-group rather than the Romans. You even use the term 'took over'.
But they didn't took over as a "dominant nation", but as a what remained of the western Roman state during and after its collapse. I wouldn't say replaced, because it wasn't made smoothly, but it's not the take over of an anachronical nation, or of a political faction whom networking would be somehow distinct from Roman institutions.
I'm not sure I agree with you here (the way that such a state could form is innumerable, if each are unlikely/never happened), but I think that comes to a point of philosophical differences when it comes to alt-history.
I could be convinced otherwise, but I don't see a reasonable way for the early VIIth century exarchate to evolve as a Romano-Berber roughly unified state. It's not philosophical, tough, than trying to understand the historical situation,
from what I could find as secondary sources.
If the forms it could take are innumerable, I'd be sincerely interested to know which
practical forms you're thinking about.
Define a strong institution.
Roughly, a relatively imprevious organisation, that while evolving, can either self-sustain itself, or can be guaranteed its stable sustain.
For exemple, the democratic institution in America or France is relatively strong, while democratic institution in Belarus or Gabon is not. Or, for what matter Britain, the militia institution wasn't strong as it greatly declined if not collapsed as soon the Roman state disappeared from Britain while it survived elsewhere, if weakened.
Because I'm not sure I agree that you need an institution of a significant strength to say if something can be a 'Roman State', as long as those who are fundamentally in charge, consider themselves Roman.
Don't forget that what defined a roman state wasn't just the self-perception (otherwise, we would be forced to consider the merovingian duchies of Aquitaine as "Roman states" which, obviously, makes little sense), but a political identity and relationship to roman institutions.
A Britton high-kingship, while including several roman or romanized features, would be a departure from roman institutions that were dependent on imperial sustain (notably trough military presence). Assuming that Brittons would define themselves as Romans as for their primary identity (which would already be a debatable assumption, as it seems they stuck to tribal/civitate identities, with the territory undergoing important break-ups of break-ups), while building-up on what remained of these.
The point is less to depict sub-Roman Britain as "unRoman", but to point that for all the cultural and sociel creolization that resulted from centuries of Roman rule, the imperial institutions simply didn't fared well without the military and administrative imperial presence : it's no wonder if the first Anglo-Saxon ensemble to emerge from the tribal mix-up of the VIth century were kingdoms in relation with Frankish courts and trade trough Gaul (Kent, East-Anglia, Wessex, mostly).
Eventually, it's not about needing institutions of significant strength to be considered as a "Roman state" : it's about the survival of these institutions (even if weakened) on the long run to allow a Roman identity to be maintained.
My understanding that the Saxons mainly ignored the Roman institutions that existed as soon as they became inconvenient, in contrast to your description of the Franks et al.
It's rather complicated :
the late Roman structures in Britain were certainly less present to begin with, compared to other western provinces and regions (exceptions made of places as the aformentioned Illyricum, altough northern Spain could easily count with the caveat it wasn't as much of a military region) with the attested survival of pre-Roman tribal identities at its rough periphery (Votodinii, Dumonii, Silures, etc.) that even with the best will in the world, I can't spot in Gaul or Spain.
Does that means that sub-Roman Britain wasn't romanized? No, but it was rather more of a creolization that what happened in the aformentioned region, with relatively less develloped structures that suffered from the
gradual but quick withdrawal of the Roman army that ammounted to a collapse of Roman state for the island.
That alone wouldn't have been as destructuring, if the roman state didn't collapsed in the mainland as well : its destructuring effects were felt up to Scandinavia,
which fragmented into smaller, impoverished and
warlike entities which eventually pushed more North Sea peoples South and West, making it even destructuring for Britain.
The Saxon takeover took place in a situation where roman institutions and structures of a
really divided Britain were either collapsing or significantly more weakened than in the mainland, so Saxons didn't as much ignored the current structures (as their kingdoms generally mirrored what existed before) that probably didn't encountered much of these (safe for Saxons from the southern shores, which were settled since the IIIrd centuries, and that might have joined up with local Brittons, such as for the
emergence of Wessex).
I don't think it means Britto-Romans were doomed politically, but post-imperial structures were simply too weak and too unequally present IMO to support an unified state in the Vth century, while I could likely see the establishment of Britto-Roman (supported, rather than weakened with a good enough PoD, by Barbarians) decentralized ensembles. With time, it could evolve from something out of the
chiefdom/complex chiefdom/state formation that, among other exemples, Anglo-Saxons kingdoms went trough and possibly earlier, eventually ending as one or two coherent polities. But I don't see it happening in the VIth or VIIth century myself.
I don't claim to be vested with perfect knowledge of the period, far from it, and I least try (do I really suceed is up to anyone but me point of view

) to accept being wrong : but with all due respect, I find the "History is ASB so everything can happen" argument to be somehwat lazy and eventually preventing any discussion (as you said, the fun is trying how to manage to pull a TL within the constraints of realism).
Now it's my conception of non-ASB AH, so maybe we'll have to agree to disagree there.
I could certainly see a united Britain with a measure of good luck and half a dozen hundred years.
With such time length it could admittedlt goes in many different directions (my argumentation is mostly about forseeable and "predictable", or rather deductible, outcomes of some PoD(s). That said, with such time length, I don't think it would be considered as Roman, as we don't consider UK as an Anglo-Norman kingdom, so I must admit I'm not sure about this point.