Which would still put the local government out of the loop. When you have a currency crisis like Zimbabwe did, it kind of make sense, they would eventually abandon their own and allow foreign ones instead. But it was not without deep reluctance, considering that most transactions in Zimbabwe is completely out of its control.
Of course, we are not talking about hyperinflation in Austria. Something that forced Zimbabwe to take a momentous decision. Why would the US Federal Reserve consider the interest of Austrians - in this fantasy scenario where they go cashless, and people start using USD - when valuing their currency?
I mean, if cash transactions are in different currency than that of your local jurisdiction, then - for a lot of people - it would start to make sense to simply keep their bank accounts in the same currency (and private banks would surely offer this). This could have an effect on people who doesn't use cash at all, since they would at times deal with people who do sometimes. And then having their money in that foreign currency too is an advantage.
Of course, I am only speculating. Since we have no experience with what happens if a society abandons cash, it will be hard to tell what the broader effects of that will be. Could be a total loss of monetary control for the government, or merely an inconvenience for foreign tourists.