On the other hand they pass a lot of truly relevant laws like a new one concerning the minimum distance of chickens in egg-farming plants in 2012: just in the middle of the crisis, Greece, Ireland &c.
We are so happy here...
The tension inherent in that set-up is between the individual powers of sovereign nations and the collective policy of the group. In this instance, there's no doubt that Sweden and the UK are helping the US in vetoing, but there's a parallel concern that the EU does not have competence to negotiate on this, and I imagine a strong desire to prevent mission creep.
As to the broader question of ungovernability - I'd argue that the really interesting question is whether it is possible to maintain a firewall between economic policy and everything else, which is what arguably distinguishes the EU from other forms of weak federalism. Even before the Euro crisis there were edge cases where economic jurisdiction bled into, for example, social issues (the ECJ ended up having to effectively rule on Irish abortion law under the free movement of services). Now it's even worse. Political parties in Greece and Ireland, for instance, have been co-opted into austerity policy by having the gun of exiting the Euro held to their heads. At that point the questions of the EU's democratic deficit and the sovereignty of its smaller members becomes more pressing. At some point, I suspect, the EU membership (Euro members especially) will find itself forced to pick between actual federalism and supra-nationalism.