“But, when the fan becomes soiled gold can’t be eaten, worn or used to heat your house.”
No, but it sure as heck can be used to back fiat money, and in the long-run that’s just as important.
No, fiat money isn't backed by anything.
You're describing a fractional reserve currency, such as what we were circulating in 1955-1965, when e.g. silver certificates were still freely exchangeable for silver "cartwheels" at the Federal Reserve.