I would throw in the failure to deal with the economic impacts of their single unified currency. The eurozone ties the poorer states to the richer states, they serve as a forced export market and cannot depreciate their currencies to provide labor-intensive products/services. They clearly need capital transfers to shore up the budgets of the poorer states (or direct spending to relieve some of the basic burdens of infrastructure/medical care/etc). The US accomplishes this with Federal spending - the coastal states have booming economies that pay much more in Federal taxes than they received in Federal spending; that money goes to the poorer states in the heartland and midwest.
You don't act like a single nation either. When a hurricane hits the US we don't worry about who foots the bill, we collectively respond at the Federal level to solve the problem, in the expectation that if we ourselves have a similar disaster then everyone will step in for that too. You can't say that about, say, the refugee crisis in the EU, which has entirely been lumped onto the backs of the states least able to afford to deal with it.
Basically the EU's problems are fundamentally the same as those of the Articles of Confederation. You can run down the list: no Federal military force, no Federal ability to tax and spend, a body of members who espouse values of unity but actually stand for the values of their own independence and sovereignty, who work against each other every step of the way if it'll mean a bigger slice of the pie for themselves, etc. Fundamentally you still view yourselves as sovereign nations and national citizens, rather than Euro Member States and Euro Citizens.
Honestly what the EU really comes down to is a bunch of tiny, insignificant states who think they deserve a seat at the grown-ups table of international politics but are unwilling to make the sacrifices they need to get there. You don't have one foreign policy, you have 28 foreign policies, and the same thing internally.
The EU is first and foremost an experiment in integration of economic and foreign policy, it clearly doesn't work at that. If you want to be a United States of Europe, then you need a Federal government with some authority. If all you want is European Peace, can't that be handled at the United Nations?