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.

I think they did? At the cost of very slow recovery, which I think is the problem with austerity measures.

Austerity is not a "problem", it's simply the only remaining option when you run out of other people's money.

Countries can inflate their debt away at the cost of eroding savings and new borrowing becoming more expensive, no?

https://www.statbureau.org/en/greece/inflation-charts-yearly

Countries with poor credit struggle to issue debt in their own currency. Being able to inflate your debts away is a privilege of the rich.

Countries can't push that approach very far before new borrowing doesn't just become more expensive, it becomes effectively impossible.