Exactly, and the most popular religion in the world is basically cultural continuity with late western Roman culture and its largest most organized official form even uses the Latin language and maintains political institutions with continuous lineage back to Rome.
I don't think there's a real timeline where Europeans were wandering around the ruins of the suddenly collapsed empire wondering who these people were. They knew fully, but the economic and political (and physical! plague!) situation meant they were unable to reconstruct the Pax Romana.
But they certainly tried, even when they were former "outsiders" like the Franks (who, in the west half of their realm, even dropped their native tongue to speak a form of Latin!).
In fact I feel it's possible that actual "progress" wasn't possible in Europe until philosophical thought moved beyond late-Roman neo-Platonic & neo-Aristotelian boundaries, and into new radical enlightenment era concepts.
Knowing fully what happened to Rome, if it’s even possible, would have been limited to the educated class. Stories would still have come out of the peasant class, whose folk memory of Rome couldn’t match what the clergy had.
Eventually we did remake the Pax Romana, but it took two utterly ruinous wars, humanism, and a working knowledge of free trade to recreate the system that Rome had managed to create almost by accident.
What has been created with modern global capitalism is extremely different from the Roman form of commerce and administration. Not just in technology but in complete ideological foundations. Not only was slavery fundamental to its whole foundation, its accepted and normal practices for doing business would today, in the west, be considered forms of absolute extreme corruption, and their justice system would absolutely horrify us.
I'd posit that today's world is what it is because it is partially on built enlightenment values that stepped away from the old Roman model, not back to it.
If you look at what the Romans in the context of their own time, it looks much different. Citizenship was an open class, unlike most everyone in the Mediterranean. Slaves actually had rights. Conquered peoples were able to earn tributes in future conquests. Women could own property. Yes, it’s all terrible by our standards, but becoming Roman could actually improve the lives of subject common people.
Reproducing the previous society is the mistake, because it wasn’t just the society. It was the free trade and lack of Mediterranean piracy that brought prosperity. The relatively increased civil rights that brought stability, and Senate that brought legitimacy. They had a model that produced the best economy and civil rights for their era, but looking backwards to past glories can’t replicate that.
Three of the 10 most common languages on Earth boil down to "Latin with a bad accent".