Desperate I think actually it's worked pretty well, on the whole, if the aim is to prevent war and to improve people's living standards over a timeframe of decades and half centuries.
The refugee crisis is a hard one because there is no easy answer. There's a reason there are economic migrants and refugees coming here, it's because it's a safer and better place to live with more opportunities than the places those people are coming from. When you have people drowning in boats and sneaking across rivers and in refrigerated lorries you cannot magic them away by not being part of the EU. Honestly the solution should be to make the places that people already are, safer and with more opportunities, so people won't come. The border states of the EU are some of the least equipped to deal with large numbers of migrants and refugees and yet they come and wash up on beaches and walk across borders. I don't think there is an easy fair solution to this problem except to work together (perhaps through an elected parliament and executive branch where each country sends elected delegates?) and each take a fair share of the problem, whether financially or resettlement or both. And long term, try to get people to want to stay at home by improving security and economic opportunity through international aid and trade.
Greece is another hard one because the effects of one country's financial mismanagement affects us all, not just in the EU but globally. Look at the Asian financial crisis caused by the Thai bhat collapse, Russia's bond defaults in 1998, the credit crunch in 2008 which really was the US and UK causing contagion elsewhere, Argentina in 2014. Greece has two main sectors of the economy, shipping and tourism, which were absolutely nailed by the 2008 crisis and it hit a country that already had huge debt and huge public deficits. They couldn't devalue their currency when the money stopped flowing in (via government debt) because they were part of the Euro. I think it was massively mismanaged because Germany was the largest creditor to Greece and really was their enabler - but that means next time, do better - don't let Germany have such a disproportionate role in internal EU debt, or to have such sway in the ECB, etc. These are lessons to be learned and rules to be changed and tightened.
Fundamentally, we humans, governments, bureaucracies learn by fucking up and trying to fix them. I refuse to say the EU as a whole is a failure because the rules and procedures, when tested, didn't work perfectly or even at all. That means we improve and do better next time. The history of Europe and our place in it is long and very very bloody, and we have a lot of collective challenges facing us like climate change - no one country can tackle that on its own and as a bloc we are stronger and can bring the world along with us. I really think the EU project is better with us in it and we are better in it not out.