And conversely, that the transition from Biblical Israel (confederation) to a united kingdom under Saul (federation) is mirrored by the U.S. moving from separate states under the articles of confederation to a centralized government with Washington as its first president?
Or am I just an idiot?
One can make the comparison of the two in a very broad sense, but beyond that is where the similarities end. For starters the United States, even in it's post-revolution infancy was and is many times the size of Israel. The entire levantine coastline, including many more areas than simply that of biblical Israel, from modern day Israel's Red Sea port of Eilat past Antioch in modern day Turkey is under 600 miles whereas the 13 colonies (roughly Georgia through to New Hampshire) would have been a distance of over 1100 miles.
Aside from these obvious size differences which are considerable and would impact trade, communication and governance one has to take into account the fact that life was very, very different between the two. Israel in 11th century BCE would have more or less a tribal confederation of various Hebrew tribes of farmers and herders who likely would not have had a cohesive sense of a state and might have only very loosely thought of themselves as being the same people. The archeological record seems to show that the presumed date of this unification was much earlier than when any unified polity existed in Israel.
As for the comparison of Washington to Saul I can't speak too much to that other than that Washington's existence is attested to by many sources of the time, including writings of his own, whereas Saul has little verifiable record of ever existing.