Although this is far from uncontested, a lot of historians would be quite happy to say that 'Roman civilisation' did not collapse, even in the west. Instead, it experienced a series of transformations; this debate is laid out quite well by /u/iguana_on_a_stick here. The view that I favour is that the 'fall' ought to be summarised as the collapse of centralised governmental infrastructure, that is, direct rule from Italy. Roman institutions in the provinces (or within Italy itself) however did not collapse. I explained this process recently here. In short, the average Romans' way of life did not go away and the Western Roman Empire was not murdered, but had instead, in the apt words of Guy Halsall, 'accidentally committed suicide'. After all, no-one had actually set out to bring the whole empire down, but had instead sought to find a more privileged place for themselves within the imperial superstructure.

What about contemporaries then? I provided a broad overview, with some extracts, here, but it is a bit basic, as I intended it to be more of a list of recommended reading instead. I can expand on this in more detail though if anyone has any follow-up questions, but I think it provides a reasonable summary of the attitudes held by contemporary Romans. More telling sources from Italy and Gaul from after 476, a date of very little consequence, are analysed here. I also focused on Rome itself in this answer, since the city of Rome after 476 provides some of the clearest evidence that 'Roman civilisation' did not end. Indeed, it would be the plague and war of the 540s, a conflict that was ironically started by the eastern emperor Justinian, that is now frequently seen to be the most decisive in ending this longstanding cultural continuity across the 'post-Roman' west.