This is not something particular to America. Europeans have always stressed the continuity from antiquity. The pope uses the title "Pontifex Maximus" which was one held by Ceasar. Emperors (like the Holy Roman Emperor and the Zar) claimed to be the legitimate emperor of Rome. The Enlightenment thinkers were also heavily inspired by Rome and Greece, although preferred republican Rome to the time of the emperors.

The reason we know so much about Rome and Greece is that we have a lot of written material - histories, literature, poetry, letters, laws etc. Archeology is a relatively new science, while history (the study of written accounts) have existed since antiquity.

History is not just a question of what happened, but also about the effect of what happened. For example the Egyptian or Sumerian literature and philosophy might be just as magnificent as the Roman, but since those languages were lost it simply did not have the same effect on subsequent culture as Rome had.