The United States is a country created on ledger-sized newsprint. If the founding documents still exist, the country still exists, regardless of what people are dead and what people aren't. If the founding documents of individual states still exist, then those states are still United states.
If none of these documents exist, then those that survive can reconstitute the nation in whatever way they please by creating new ones, which was done in Charleston. If those that recreated the country want to call their chief executive President and their country United States, then that's their business until somebody with bigger guns and balls tells 'em different.