> I, a foreign entity, have sold something to an american and now have 10 dollars and zero treasuries.

Or you sold something to a non-American entity in a dollar-based market, eg. oil. The dollars do come from America to begin with, but once they get "out there" they work as a medium of exchange for whoever wants to use them for that purpose.

> Presumably i now either repeat the cycle or use my winnings to spend on us output.

What you are missing is that these dollars can be circulated indefinitely in the global economy without ever repatriating, because they are valuable and useful as actual currency. They may never come back to the US.

Because the world trades using US dollars. Country A needs to buy something from Country B. Country A needs to buy/get dollars to buy stuff from Country B. Country B will not accept anything but dollars or gold for its products because it also needs to buy other stuff like oil in dollars from other countries.

> Presumably i now either repeat the cycle

Most of the time this is exactly (foreign or not) institutions do.

Think about it, if the 10-dollar treasury is due and you got your money back, the US debt will go down by 10 dollars. However, in our reality, the total amount of US debt almost never goes down.

Of course some interest will be used in other ways, like spending on the US goods or staying as cash to provide liquidity. But at the end of the day, the most popular way to spend the money got from due treasuries is... to buy more treasuries.