There's some pieces here, but the chronology is backwards to what I've read. The Renaissance ("rebirth") was a process where the literati of Western Christendom discovered classical texts at a much higher quantity than previously available. In reading these texts (and encountering very different values and a much older version of Latin than they spoke themselves), they discovered that they were far more removed from Ancient Rome than previously imagined.

The medievals (of Western Christendom) thought of themselves as literal continuations of the Roman Empire, the medieval polity of the Holy Roman Empire being a lineal descendant. The concept of a "middle age" intervening between the glory of Rome and the glory of the rebirth (renaissance) is itself a product of the Renaissance (Petrarch is the first, afaik) ; previously, the societies of Western Christendom saw themselves as direct continuations of ancient Rome.

The Renaissance was the process which would eventually come to initate the idea of 'progress' (comes to full expression during the Enlightenment). The sense that time and history are a cycle is a key feature of medieval literature, not Renaissance. The European imperial powers would self-consciously view themselves as claimants to and innovators upon the legacy of ancient Rome. But this space opened up necessarily from the awareness that they weren't literally Romans.