It’s so nice to feel useful. 
Not only medieval Greeks and Italians. Granted, the average Joe in the middle ages didn’t journey very far afield, but if you DID travel, where did you go? If you were clergy, you went to Rome. If you were going on a Crusade, or trading along any of the routes between Venice and Constantinople, you would most likely pass through Greece, or the Hellenic cities in Asia minor. And you brought stuff home with you – stories or descriptions or actual things. There are Dutch woodcuts of the Seven Wonders of the Ancient World from the early 16th century – 5 of the seven are Greek or Greek-related.
This fits in with your explanation above – that the “enlightened” thinkers of the 18th century were looking to Greece and Rome, and this was the same philosophy that was driving the people who decided what it meant to be educated and cultured in the US. Again, most did not travel, but sometimes the wealthy and privileged did. They brought back souvenirs of their travels. There is a particular style of oil painting that depicts ancient ruins and other famous sites – these paintings were practically mass produced, and sold as souvenirs. Even at the time, they were not usually considered great works of art, but rather held the same function that our snapshots do today. They were reminders of your travels, and reminders to other people that you were wealthy and aristocratic enough that you, or a member of your family, could travel. Plantation and other estate homes from the early 19th century are lousy with these paintings.
Later in the 19th century, a galvanizing event occurred – Heinrich Schliemann discovered the ruins of Troy. Putting issues of scholarship aside, the general public was suddenly excited by the fact that Troy was a REAL PLACE. The ancient Greeks had been drifting around the popular imagination for a while, and perhaps they would have eventually been replaced by a fascination with another culture, but this discovery was significant because it created a focus and a new relevance for Greek mythology.
Enter Neoclassicism … you have banks, libraries, and public offices built to look like Greek temples (there has always been a strong vein of this in American architecture, but now it explodes). Museums and cultural institutions are in a period of expansion, and they are judged on the quality of their antiquities (those Greek pots again). There is probably some trickle down classicism at play here – scholars make more discoveries in this field, university collections and museums get more actual stuff to display, kids get dragged to the museum, where some of them at least get interested in the larger-than-life statue of a guy cutting off a chick’s head (and she’s got SNAKES for HAIR, how cool is THAT?).
This brings us up to the present day, and why most of us had a unit in school where we learned that Athens invented democracy but America made it better, never carry a fox inside your shirt, and beware of Greeks bearing gifts.