Network can work pretty well at $10/BTC and at $10000/BTC. Bitcoin is "backed" in the same sense as gold: it's just unforgeable, divisible scarce asset which makes it a good collectible. If it has any value at all (in the eyes of some collectors), it can be used as a medium of exchange and can potentially acquire many more collectors willing to hold it in expectation that they can sell it any time for something else valuable.
USD is a collectible too, without any "backing". Some time ago USD was gold backed, meaning it was a promise to give you a certain amount of metal from the bank. Since 1972 (actually, for average Joe it all started even earlier, at Bretton Woods agreement) USD is officially not backed by gold. It's a pure collectible protected from forgery by law enforcement (both paper bills and banking transactions) and also protected from competing collectibles by various regulations and controls. As long as govt does not print dollars too quickly, it remains being a scarce asset which can be used as a collectible. Enforced requirement to pay taxes only works as long as dollars are demanded on the open market as a scarce collectible. If dollars are only needed as tokens to be purchased from govt with real goods (or bitcoins) to be paid back to it, then they have no meaning as money and it's going to be just a fancy way to pay taxes in something else.