What would be a plausible reason for why the Founding Fathers Tribe would want to rebuild the USA?
They think that the USA was a nice place, and they want to make something just as nice.
There is plenty of examples in literature. My favorite ones are all concentrated in one single videogame - Fallout: New Vegas. In this game, there are some tribes that wish to emulate, or even surpass, past civilizations in terms of greatness:
Caesar's Legion is a group of people who cosplay as legionaires - think roman centurions carrying laser rifles. They have a very poor and limited grasp of latin but try to shove it into their day-to-day communication anyway, from salutes to how they name themselves. They implemented a currency based on coins with the face of Caesar in it. They do it for no other reason than thinking that ancient rome was the epithome of cool. The fact their slave population will rebel here and there does not seem to bother them, and the doctrine of trying to annex neighbours through war has made it really hard for them to make friends, but still...
The New California Republic is kinda trying to rebuild California. All the way to the flag, though now it has a two-headed bear in it (because radiation). They try hard to look like the good guys because, you know, freedom and democracy (the Legion also claims this about themselves). At the end of the day they are suffocating people with inept leadership and anacronisms, while thinking they are the bastion of all that is good.
The Khans want to become the mongols of the 23rd century. They don't know fertilizant about who Ghenghis Khan was other than knowing that A) he went to wars and B) his helmet had horns. But hey, rule of cool, right?
The Kings are the only ones who are even close to going the right direction towards good and enlightment. In the ruins of Las Vegas they found buildings and memorabilia dedicated to a man who was called "The King" in his own time. They don't know much about Elvis Presley - they don't even know his true name - but the leader of the Kings concluded that if so many people worshipped the man, then he was some kind of holy person in his own time, and by having his tribe emulate Elvis the world become a better place. Yes, this is a cargo cult that seems to have come out of a certain Mojo Nixon song.
What all the tribes above have in common is a very limited knowledge of the past, oftern fragmented and distorted. But the knowledge they have places whatever they are trying to emulate as the best thing ever. In some cases we, from the real 21st century, might be complied to agree based on our own biases. In most cases, though, such adoration of "a time that was better than now" defies logic and disencourages further research.
Back to your guys. One thousand individuals are not enough to rebuild the whole of what makes the United States of America. What era from the US are they trying to reenact? The 1800's? The 20's? The 70's?
Also notice that, unlike the majority of countries in the world, the US are huge. It's a continent-sized country with more than a couple hundred million people in it. There are many cultures, and cultures within cultures across the states. Not seldomly their ideals clash due to their antagonistic schools of thought. One thousand people are probably not enough to represent them all. Are they really going to cover all cultural groups that make America? The yuppies, baby boomers, generation Y, generation Z, millenials, hipsters, hill billies, WASP's, B-boys, hippies, emos, jenova witnesses, dreamers, flat earthers, vegans, alt-righters, antifas, militant atheists, soccer moms, metalheads... And these are just a few. Are all of them going to be represented in the new, 26th century USA? I think not.
Those people probably saw a very limited snapshot of a small group of americans from a short span of time and concluded that the ideals of those people from the past matched their own tribe's ideals. Then they performed the following silogism:
- USA was great;
- Those few people from the past that we identify with lived in the USA;
- Therefore, if we emulate that very limited sample of people from the past, our new country will be as great as the USA were.
Which makes for a cargo cult at best. Whether this idea succeeds into doing a future nation that thrives, though, is up to your writing.