In fact, even Rome built its public buildings and spaces to make it seem like the "new Rome," except that Rome to them was Athens or Mycenae.
There's a very direct line of descent from the White House to bronze age greek public architecture. That template for a building has meant "power" forever, and that's the symbolism everyone always goes for.
In my mind, the line goes forward, reflects and refracts through Europe, the Catholic church, the Renaissance; the Freemasons, whose first lodges in America were in Philadelphia; the "Sons of Liberty" gathering at the Green Dragon tavern - among them Paul Revere; Ben Franklin, who commissioned the design of the Great Seal of the United States, with the pyramid symbolism; and reaches George Washington, the first U.S. President and Grand Master of the Masonic Lodge in Virginia. And who were masons but builders and architects?
The morsels trace an admittedly sinuous and questionable lineage, lost and found across time and place, revived through myth and homage, and, I believe, still feeding some mad dreams of an empire.
Arguably, Rome didn't even fall when Rome fell. A Roman emperor continued to rule over a roman empire from constantinople (which had already been the capital for a long time) for another millenium.
Western intellectuals just had an obsession with "the fall of rome" as a concept. Lots of historians studied tomes of the stuff. Everything can be projected onto it: social puritanism, christian puritanism, anti-christianity (they abandoned jupiter), anti-immigration sentiment, anti-german sentiment, etc.