That problem with you, is your so reflexively focused on government debt that you can't get your head around the fact that most of the Southern countries in trouble DIDN'T HAVE LARGE GOVERNMENT DEFICITS. They had large and growing public debt most of it related to borrowing to buy private assets. And unsupportable wage inflation.

Go look it up, the governments of Italy, Spain, etc except for Greece weren't borrowing beyond their means. What they were suffering from was out of control inflows of cash. Inflows of cash in a country with it's own currency tends to be self limiting, because as money flows in the exchange rate adjusts making investments less attractive. (A country with a well run central bank will actively adjust the exchange rate to prevent the inevitable panic and flight that happens at the end of a bubble, or you have a bubble and it sucks for a while)

Part of the problem with the Euro is weaker member countries can't control the flow of capital into and out of the country. Yet they are on the hook for the mal effects that result.

This is wrong.