The dollar is essentially backed by oil and the fact that it's needed to trade with the largest economy in the world, not to mention the country with the largest military.
The fundamental source of demand is the requirement that US Taxes must be paid in dollars. The utility of using dollars to trade with the US economy is simply a consequence of this fact.
(1) the infaliabilty of its public ledger (which means it is especially useful where trust is low and where trust is needed).
(2) The ease of making a transaction.*
*Aside: since the bitcoin transaction volume currently exceeds the limits it was specifically designed to handle, that means that transactions currently either cost a lot of money or take a long time to be confirmed. But those are issues that could be resolved either with lightning network (or other off-chain solutions), side-chains, forks (soft and hard), or even by a whole new variation of the blockchain concept. Although bitcoin transactions admittedly currently aren't cheap or fast doesn't mean blockchain necessarily will always be like that.
Except when they roll it back because people with more money than you demanded it after they got screwed somehow.
The downside is the US government's outlay is larger than it's income, but in practice that's less important than you might think.
Let's say I think dollars are worthless, and my whole community works off something with real value, like shiny rocks. I'll still have to pay the IRS next April. They may take shiny rocks, but instead I could trade just one or two rocks for a big pile of worthless dollars and get Uncle Sam of my back for next to nothing. Unfortunately for me, every other shiny rock user in town is going to have the same idea, so they'll bid up the price of dollars until trading rocks for dollars doesn't feel any better than just paying my taxes in rocks directly.
The only faith that "full faith and credit" requires is faith that I'll have to keep paying taxes.
Taxes are why I think a piece of paper with a bad portrait of George Washington has some intrinsic value to me, whether or not there's a secondary market for them. It's why I think the purchasing power of that piece of paper is anchored to a real phenomenon in the real world, and why I don't expect that purchasing power to move around much.
Think of it like this. If the dollar lost 20% of it's purchasing power overnight, I would be pretty damn sure something important has happened in American governance. Bitcoin just moved 20% because...?
The last time the dollar was backed by any asset was when it was under the gold standard. Boy was that a mistake.
No, the petrodollar system effectively ended with the 2008 financial crisis[1]. The US dollar is just backed by debt now.
EDIT(wrong URL): https://uk.reuters.com/article/uk-usa-bonds-petrodollars/com...