The European Council has stumbled from summit to summit over the course of this crisis, and the European Parliament, despite being the most democratically legitimate EU institution, has not been the site of the central debates about how to fix the Eurozone. Nosemonkey has just written a great article on the need for more democratic legitimacy in the EU, and the ability to finally make a decision. However, I disagree with him - and the German Christian Democrats - that a directly elected president is the answer. I don't think a presidency could work in practice because Europe's soul is distinctly parliamentary.
Out of the 27 (and soon to be 28) Member States, all but France and Romania are either parliamentary republics, or constitutional monarchies with a parliamentary system. Even France and Romania are only semi-presidential. Parliamentarianism is part of the political culture of the Member States, and part of European political culture generally. In the EU system, the European Parliament elects the Commission and the Commission President - who is nominated by the European Council on the basis of which Group in the Parliament won the election, which is reminiscent of the role of the monarch in many European constitutional monarchies.
Of course, the declining election turnout for the Parliament has caused a crisis of legitimacy for the institution. Without a clear platform for electing an executive, and because European elections are second-order elections, voters have few clear political options on the European stage. Europarty primaries and common campaigns centred around a Commission presidential candidate (read: prime ministerial) might eventually change that, but it's a long hard slog and is unlikely to give the boost in legitimacy needed overnight. In contrast to this, an elected president, usually seen as a directly elected European Council president or a combination of this role and the Commission President, is a relatively simple institutional tweak, a small treaty change, and it would establish a clear leadership position for election.
I don't think it would work culturally or institutionally.
Culturally, a presidential system would present too clear of a binary choice in a continent used to coalition-building on the national level. I don't necessarily just mean the Member States used to coalition government (though many of them are), since even in parliamentary systems where there is a single-party government, parties tend to be broad churches. It also provides for a strong government, which I believe is culturally embedded across the continent, but not government that is simply built around one person, despite the growing presidentialisation of the parliamentary system (France is a bit of an exception here, although with a strong president, there is a strong executive). Parliaments provide a space for ideological debate between political factions, and a government based on a coalition of support across a majority of society as represented in parliament. While a directly elected president may have a strong mandate on paper, I think that this would be culturally undermined by the suspicion that the presidency's interests and loyalties are too narrow, and not sufficiently representative.
A presidency would also be quite weak institutionally, since he or she would sit in the European Council as head of an institution in which he or she had no power base. Sure, whoever is elected would have greater legitimacy than the national leaders in the context of the European Council, but those leaders represent important and valid electoral interests, and will not - and should not - simply bow to a European President. The president would also lack a power base in the European Parliament, and could be politically different to the Parliament's majority. It's hard enough for cohabitation to work in France or the US without trying at the same time to build up a profile and confidence in an elected EU president. Without a natural power base around which to build coalitions or majorities for proposals, the presidency would likely dash electoral hopes raised in the election due to the inability to honour his or her manifesto. This goes back to my point about the culture for strong government: weak elected governments rarely win a second term, institutionally weak systems are rarely respected. And who would vote for an institution that cannot deliver? Is that not the argument for the declining European election turnout?
I have the personal bias of believing in parliamentarianism, but it seems to me to be the only system that could possibly work on the European level. There needs to be a broad coalition which a majority can not just buy into at the time of a momentary choice between two candidates, but generally as well. This is best done by a parliamentary coalition comprising of a majority of opinions, not a single individual. The Commission could evolve into a cabinet with a Prime Minister figure at its head.
A lot of time and effort would need to be made to even attempt to deliver this, but a presidential system is the same old attitude of tinkering with the institutional set-up when you get down to it: real work needs to be done at the local level, and only political movements - parties - running for election on common platforms can provide the incentive and promise realistic results in order to sustain this work. Which is why I believe a working Europe could only ever be a parliamentary one.