Arguably, Rome didn't even fall when Rome fell. A Roman emperor continued to rule over a roman empire from constantinople (which had already been the capital for a long time) for another millenium.
Western intellectuals just had an obsession with "the fall of rome" as a concept. Lots of historians studied tomes of the stuff. Everything can be projected onto it: social puritanism, christian puritanism, anti-christianity (they abandoned jupiter), anti-immigration sentiment, anti-german sentiment, etc.