Exactly, they don't give you dollars, they give you a dollar denominated liability.
The contention upthread was that foreign banks could do the same thing to print dollars. And, again, it's complicated. They sorta can, but only in the sense of placing a gargantuan short bet on the dollar. (A ton of short selling on a small security can push the price down for the same reason). So no one does that, because no bank wants to play that kind of game (or is allowed: all those large foreign banks are themselves regulated by their own national regimes).