The US special role is based on two things, size and stability. Everything else is icing.
I don’t think you can un-pick dollar stability from its desirability. The value of fiat currencies is based solely on confidence, nothing else. You simply can’t say a currency is stable because people are confident in it, or that they are confident in it because it is stable. They are both different ways of saying the same thing. We measure confidence in a currency by analysing how stable it is in the market.
Right up until the point you find a US citizen anywhere in the world with a bill from the IRS. Then you find the true nature of the demand for US dollars - you can always sell them to an American with the IRS breathing down their neck.
If they exchange the dollars with the euro then someone in the EU will end up with those dollars and that person can't do anything but buy US products.
Anyways, my point is that Eurodollars can be created, passed around among foreign entities, and destroyed without ever necessarily touching the US flow of funds.
That's not how any of this works. It is far more common to price things you buy from the US in USD. Which is also why monetary policy works.
Half of exports from the eurozone is in EUR. About a quarter is in USD.
> Why are these American dollars valuable to you?
You need them to service your dollar-denominated debt or to put them in your reserves.
Imagine you're a foreign country. You've put in a bunch of labor and manpower to make goods for the US. You sell them to the US, and they pay you with green pieces of paper.
Now what? Why are these American dollars valuable to you?
Because you can give them to other countries and they'll give you goods in return? Then why are these American dollars valuable to those other countries?
Because it's turtles all the way down?
I think it's because somewhere along the line. Some country will take those dollars and buy a F35. Or a IP license from Intel. Or the rights to distribute the latest Marvel movie in their country. What do these things have in common? No other country in the world can produce a equally good substitute.
That's the core of what still gives the USD its value. The US being able to produce goods and services that the rest of the world has no substitute for. And because the rest of the world must have USDs to buy American technology and expects this to continue into the future, this allows the US to consume more from the world and have a large consumer market.