The problem with this question is the assumption of 476 AD as an important date. While a lot of historians will generally accept this as the fall of the Western Roman Empire, and the general public uses this as the benchmark for teaching the fall of the Western Roman Empire, the truth is slightly more complicated.

While I can't speak in depth to the minutiae of the social aspects of this question, I can give an overall answer to the question. If someone who has more in depth knowledge on social attitudes of the time, I'd love to read that as well.

First, Dark Ages is a misnomer and an inaccurate assessment of the period immediately following the fall of the WRE. A lot of work has been done here and over at /r/BadHistory on this. Check out some of that stuff if you are interested in learning more about that.

There are a few points that must be remembered in the centuries leading up to 476. By the early 300's, Constantine the Great had essentially solidified the shift in Roman focus to the east that had begun with earlier emperors. The west was no longer the primary concern in some ways. Rome, while the emotional center or the "spirit" of the empire was no longer the center of the empire politically in many ways.

It must also be remembered that by the time the 400's had rolled around, the west was increasingly "barbarian" in make up and less "Roman" if I can use the terms loosely. The Western Roman Emperors were increasingly reliant on generals from the incoming tribes. It was at this point that the borders were breached for good, as Alaric and his Gothic tribes (who had once been loyal to and fighting for Rome) sacked Rome in 410 after several campaigns and then settled in Gaul. Honestly, the sack of Rome in 410 would have been much more of a psychological issue and a "the world is crumbling" moment for the people of the western empire and the Italian peninsula in particular.

As the 400's progressed, the Eastern Empire tried to set up some sort of stable government, but invasions and incursions from Vandals, Huns, and other moving tribes threw the entire western empire into something that could be described as organized chaos. When the last "Roman" emperor, Romulus Augustulus was deposed in 476 and the Emperors seal sent to Constantinople.

While there were still claims to supremacy by the Eastern Empire and pockets of recognized Roman control in small portions of the empire (such as Nepos in Dalmatia and Syragius in Gaul) Roman control of the west was essentially over. While there would be attempts to retake the west, particularly from Justinian, the western empire would never be the same.

Now to answer more specifically your question: I don't think people in the Roman empire thought" this is the moment Rome is finished." While they may have understood the importance of events such as the deposition of Romulus Augustulus and setting up of "barbarian" kingdoms, you have to remember that they have been living through warfare, upheaval, invasions, and conflict for almost a hundred years or more. These events in and around 476 were just more events in a long trail of events that would continue throughout the medieval ages. It is a uniquely hindsight process that allows us to say "this is when things changed" and sometimes we even get it wrong.

Hope this helps, feel free to ask questions if you want something clarified!

Sources:

basic google search sources

"Decline and Fall" from Gibbon

"A Short History of Byzantium" John Julius Norwich

"End of Empire" Christopher Kelly

"The Grand Strategy of the Roman Empire" Luttwak

College courses on:

Byzantium

History of Christianity

Middle Ages