If the U.S. government fell, any bodies of power that arose in the aftermath would be quite likely to repudiate the debts of the old government. So no, the dollar isn't quite backed by the material value of everything in the U.S.
It is backed by the stability of the U.S. government though.
True, though it won't fall in an instant, so there's some warning. And other governments seem likely to hold at least some power over any brand-new government, so it's still unlikely/impossible that they'll get nothing back from the debts.
There are also examples of post-fiat currencies. Immediately following the toppling of Saddam Hussein there was still economic activity denominated in the old Iraqi currency, most probably on the assumption that it would be convertible at some rate with whatever came next.
There's also a certain amount of inertia. If I stop taking your old Iraqi currency, you'll stop taking my old Iraqi currency, so it's worth a lot to me to maintain the fiction, to believe really hard that the old fiat currency is still meaningful.
It is backed by the stability of the U.S. government though.