There's a lot here to respond to, and in the end it's mostly speculation, but a few things.
In regards to the comparison between Franks and Goths and Saxons, it's not really the same. In those cases they took over the existing Roman state and continued its administration, leaving the institiutions of civil society much the same and even adopting Latin as a language and Roman cultural practices for themselves within a couple generations.
By the time the Saxons arrived en masse in what is now England the existing Roman administration had pretty much collapsed. That seems to be the archaeological evidence in my recent reading: drastic declines in the economic vibrancy of British communities. Things like amphorae and other mediterannean trade goods vanishing. Rich people burying their wealth to hide it from invaders. And, yeah, the stories told by Gildas & St Patrick, our only real contemporary accounts from this general period, seem to agree with this: decline of local British power, retreating into small cloistered communities, Saxon (and other Germanic) raiders wea continual pest even in Roman times, yes, but after the departure of the Roman administration it becoming drastically more aggrivated, the locals were incapable of defending themselves, and the British turned to Germanic mercenaries as a defense (a common Roman tactic). Which predictably backfired.
I've read two books commenting on this period of British history recently, and both disagree remarkably on the degree of cultural and physical replacement. But on the whole the genetic and archaeological and linguistic evidence seems to me to support the statements by Gildas and St Patrick: the British were on the whole replaced & expelled (to Wales, Cornwall or Brittany), or exterminated. This is reflected in part by the fact that only a few Saxon settlements carried forward on top of existing Romano-British settlements, and most adopted new names entirely. Even some prominent ones like London were left pretty much uninhabited for generations before being re-settled later.