Influenced. Not modeled on. Every republic since Cæsar has been influenced by Rome, including but not limited to its discussion on the relative benefits of monarchy, oligarchy and democracy.

There were very few republics between the fall of Rome and 1776. All basically local governments and very feeble.

Not a republic, not technically, but the English Parliament, no doubt. Bicameralism, impeachment, an executive cabinet--these did not exist in the Roman Republic.

Of course, English democratic tradition was influenced by Rome's. As was America's. But it's a stretch to say the U.S. was "modeled" on the Roman Republic. It was a new system of government. (If one were to claim it was modeled on anything, it would have to be the English government.)