Nope; if a country leaves the euro it suddenly matters which banks’s name is on your debt notes.

That's true although we haven't seen any country leave yet.

Does it? Aren't they Euro-denominated anyway?

No a Greek-issued euro debt instrument would become a new currency decoupled from the euro with a floating exchange rate if Greece left the euro, for example. That’s what all the hooplah a few years back was about.