Actually, it is backed by all the goods and services which are denominated in that fiat. The price suggested for an exchange is a confirmation of the trust the producer has in the currency. When you accept payment for your work, you do the same. If you think thoroughly about this, you will inevitably come to a conclusion that such a system doesn't differ very much from a money system where currency is backed up by some hard asset like gold...
Like many on this forum you seem not to understand how FRB actually works. If you think otherwise, please explain how a bank can have more liabilities than assets?
So all that's needed is for people to accept payment in bitcoin and it magically becomes "real"? Cool.
In short, yes. But why do you think that most people would agree to receive payments in bitcoins in the first place? The real magic would be if you could persuade them to...