So let's take a look at the arguments against the Spitzenkandidaten.
Nobody knows who he is!
Who's ever heard of Juncker? Sure, there were some head-to-head debates between Juncker and Schulz on German TV, and there was a pan-European debate between the five candidates, but very few people paid attention to that. (The European Council, by contrast, would never pick someone the public had never heard of). This attacks the democratic legitimacy of the choice. The presidential campaign didn't go as well as its supporters wanted, but it should be noted that the lack of media coverage of the candidates was the most severe in the UK (Juncker didn't campaign there like he did in other countries, and Schulz only visited Northern Ireland because the SDLP, rather than the Labour Party, is the PES party for the area).
The support for Juncker in the German media is because of the debate they had before the elections, so Juncker is seen as the legitimate winner of the elections. Likewise the "nobody knows who he is!" argument is strongest in the British media. While this argument does matter, the problem is that it cuts both ways: how will it play in Germany if the Council picked a nobody rather than Juncker, and how will it play in the UK if Juncker is picked?
The European Parliament doesn't have the right to pick the President, only the European Council does!
The European Council nominates the next Commission President, who is then put to a vote by the European Parliament, which elects the president (later on the Parliament also votes on the Commission as a whole). This argument focuses on the legal position and powers of the institutions, though it can be read both ways - yes, the European Council makes the nomination taking into account the election results (which aren't binding), but the Parliament also "elects" the Commission and Commission President. Those making the argument that the European Council should be free to ignore the election results (or at least the Spitzenkandidaten) argue that the Parliament is making a power-grab here.
Well, the Parliament is making a power-grab. But it's not an illegal one. Through its power of election (and it's power to remove a sitting Commission), the European Parliament has an effective veto over the Commission. The European Council is, of course, free to nominate someone else, but the question being fought over right now is how much it can under the current system. The point is that the Parliament can reject any nominee that the European Council proposes, and this, along with the commitment of the big Europarties to rejecting non-Spitzenkandidaten, that is forcing the Council to deal with this issue in the first place.
There are two legal rules here, but it's the political coalitions that can be built across the Parliament and Council that will decide the issue in the end.
This would make the Commission political! The Commission should be the neutral "civil service" of the EU.
It's hard to see how you can argue that the Commission is politically neutral when the Member States have been sending politicians as Commissioners since the institution began. Jacques Delors was hardly a politically neutral figure. As the Commission has a near-monopoly on the right to propose new laws (some laws can be proposed in the Council), it can't help but be political. While the European Council sets the general direction of the EU, the Commission consults on and draws up the proposals, which have to be put to a vote of the Commissioners before they're proposed. And presumably those who oppose Juncker on the basis that he doesn't fit their political agenda think that the institution is political enough that it warrants a campaign.
The introduction of the Spitzenkandidaten will make the Commission more political, in that the Parliament - or at least the Europarties in the coalition - is likely to want to see a kind of "coalition programme", at least on the priorities of the Commission. I think we'll see something like this from the Grand Coalition in the Parliament, but I doubt it will be very detailed or will end up replacing the Council's script. The Council is still a very powerful co-legislator, and the Commission and Parliament will have to agree on them for any law to be passed.
This would make the Commission too dependent on the Parliament and destroy the institutional balance of the EU.
The Commission, Parliament and Council work together as a kind of institutional triangle: they have to work together to pass any EU laws. An argument against the Parliament having such influence over the choice for Commission President is that the Commission will become overly dependent on the Parliament for its political support and direction. As I mentioned above, the Council's support is needed in any case to pass laws, so I don't think this will have too much of an impact institutionally, but it could have the effect of making the EU a bit more majoritarian.
If there is a coalition programme, for example, the Commission might be motivated to propose a law based on the fact that has a majority in the Parliament. The Council could reject this, of course, but the public nature of the proposal makes the situation more complicated, since Member States might have to think more about their position on the proposal. Consultations with the Parliament and the Council before draft laws are proposed so I can't see anything radical being proposed - but proposals might be increasingly pitched on the understanding that there's a qualified majority in the Council rather than letting some ideas die off or never even be considered due to the opposition of one or two Member States.
An interesting argument could be had here on whether this would make national governments feel less in control of, and less likely to support, the EU while not generating enough democratic legitimacy for the Union to sustain itself without the full support of the elite of the 28 member governments. It's not really something that can be answered before it's been tried, but those supporting the move towards Spitzenkandidaten should be mindful that it's not a strategy without risk.
On the other hand, sticking with the current Council-dominated approach, which tends to favour bigger countries over the small ones, isn't without risk either. The economic crisis has largely been dealt with by the European Council through summits where clearly not all Member States are equal (you only need to think of Germany's dominance in the Eurocrisis). If bringing the Commission into line with the European elections makes the Commission more independent of the Member States and more able to treat them equally, this could be a reason to support the process.