They could neither fix the problem with the policy tools a country with its own currency would have, nor did they have the automatic support from a central budget a region or state of a country would have.
They couldn't fix the problem because they were a corrupt country refusing to face their reality and which leadership decided to blame EU for not pumping more money into their pockets instead of fixing the structural problems of their rotting country.
Destroying their peoples lives with inflation so they could continue the parade would be even worse for both Greece and EU as a whole.
Greece could have fixed their problems by cutting spending and collecting more taxes (not necessarily higher taxes, but actually collecting the taxes that were owed). Currency inflation was never necessary.
No, the Greeks were cool and calculating, turning an inflating fake fiat currency into hard assets like real estate, while not working or producing anything.
The introduction of the Euro was the start signal for each country to try to inflate as much as they can for their own benefit - since the first inflator gets the best value out of the currency. Idiots are left holding paper currency worth less and less each day. Having to work harder and harder in a never ending spiral.
Absolutely. But when you inflate your own currency, it is you who have to deal with the consequences. If you inflate everyone's currency you reap all the benefits and leave everybody else with the burden of the consequences.
Inflation isn't controlled by the ECB. Money is created when banks loan out money. Whether that be to people, companies, or governments (municipalities and so on). That's where all money comes from and how the currency is inflated.
The country which by legislation or by other means makes it easier for banks to lend out money will benefit their own citizens and companies, since they get to be the first spenders of newly created money before it reduces in value. That's how national governments can use inflation for their own benefit within a common currency system. But no government involvement is necessary. If a group of people (in this case the Greeks) are willing to take on extraordinary amounts of debts compared to other groups, then we have the same results.
Remember that all money which was borrowed in the European "debt crisis" went somewhere before the loans couldn't be paid. Somebody benefitted, and the idiots were left to pay.
"Our plan to be inaccountable didn't work, so we'll try a different scheme instead".
And yes, it's a huge problem. That's why I called the situation unwinnable.