Banks generally settle their accounts by digital transfers in their central bank accounts - no physical cash involved, they're trading IOUs from the central bank of the respective currency.

nope. No way the modern economy could function if you had to lug around pallets of cash all the time. Similar to the reasoning for going off the gold standard.

The problem, of course, is that when the money is just a number on a screen, it's very difficult to ensure that the humans running the database don't magically decide they have a bigger number than they aught to. This is an unsolved problem.