Closer economic integration would definitely solve the problem of Greece and other weak economies. Every country has their weaker provinces that would have sick economies if left on their own, but they work fine because they're part of a country with an economic policy that takes care of the entire country. The US has plenty of poorer states, for example, and they get way more federal money. Every country has areas like that, and takes care of them to some degree.

But Greece has to fend for itself, and has lost the means to fend for itself, because they're not in control of their monetary police. They're unable to devalue their coin as they should. But if they were to become part of a more integrated EU economic policy, the EU would eat up their debt without blinking, and invest in Greece to fix the problem, which is currently apparently against the rules.