well, I would dispute that - in 476 AD, the transition from "we got a Caesar in Rome" to "we owe our allegiance to the Augustus in Constantinople but he don't interfere in our Kynings rule here" was seamless
The loss of Spain and Italy to the Goths was pretty much just the tax collectors giving the taxes to someone else. The culture and material life remained the same
in briton things changed, in Gaul not so much
There was no societal collapse until the gothic wars
Now if you mean the population of ROME the city itself, yes it collapsed, but that was because they basically stopped handing out free food and entertainment. So people went to the countryside. Then the gothic wars (NOBODY expects the Gothic wars!) came
And the fall in the standard of living likewise resulted in a material condition well beneath even the pre-Roman era. -- in which areas??
I've actually got the book, just pulled it off my shelf and here's a rought critique:
Ok, I've gone on my little hobby horse - sorry about that
I am thinking of a very different book here.
The Fall of Rome, Ward-Perkins
A deep dive into the archeological record, potsherds and middens, postholes and roof slates, that sort of thing.
The collapse of material culture, and broadly speaking, its utter disappearance, point to a massive population collapse across Western Europe. Given what seems to have been a cataclysmic human disaster, its certainly wasnt just a matter of switching tax collectors.