But Greek culture... yeah, that's a retcon.
The Goths in Rome explicitly modeled their state after Rome, to the point where they considered the Western half to have survived. The organization of early Frankish kingdoms is a direct result of Roman estates, as experienced administrators were all... Roman. Rinse and repeat. It's not so much that it "informed" their institutions as that dux->duke, because they were the same position, and Diocletian's dioceses obviously survived as organizational units, along with Diocletian's organization of the empire (of which feudalism is a direct result).
During the Renaissance, ideas of liberty and democracy aped the Romans, but none of this was a cargo cult, as the ideas spread in the Renaissance drew directly from Roman historical sources (including arguments for why they did/did not work) rather than any cargo cult mythology.
Roman culture, though, was heavily focused on superstition and virtue, rather than logic, and is not in any way a retread of Greek culture.
The spurts of Greek (and later Persian and Egyptian) fashion in Rome existed precisely because they found it exotic. Otherwise Roman and Greek myths link at several levels purely and simply because they shared the same root. Romans didn't really "snatch" Gods from Greece, they were already theirs. And they "enriched" their existing archetypes with new attributes from outside as the Empire grew to pacify the subjugated peoples.