The dollar has not been backed by gold for a long time. It is backed by the Federal Reserve bank by the political power of the US and of course by the GDP of the US. Like with gold prices, there is quite some flexibility in the value of a currency, but that flexibility is contained in certain intervals because of the influence and the acting of the Federal Reserve bank. As long as the US doesn't default, the dollar has a value.
I get your point and generally agree fiat is not backed by much more tangible than BTC’s math. To be fair, plenty of people trade US dollars for US political power. Lobbying, 500k speaking fees, PACs, Charity “Foundations”. You and I can’t trade in those markets because we don’t have enough dollars (or have too many scruples)
It is backed by the GDP as you can buy goods sold by US producers with dollars. Be it iPhones, beef, corn, cars, airplanes.
And you would quickly notice the US power when you tried to print your own dollars for example.
