As for clear leadership, it must be the EU parliament. You cannot put a military under inter-governmental control. Which is good, as inter-governmentalism is bad in general.
"As for clear leadership, it must be the EU parliament."
This is a misunderstanding of how leadership works.
'Parliament' cannot provide real military leadership.
The EU Parliament above all, is an effete bureaucracy incapable of 'doing' most things.
Yes - of course the political body must be responsible and direct the military objectives etc., there is no doubt there, but that is not nearly enough.
There needs to be a very well organised and experienced body military personnel who can basically form the basis of leadership, and it needs to be effective.
The US has that, the UK and France have that at least at some scale.
If the EU said 'we're going to have an Army and pool all of our budget' and then every soldier, officer and seaman were under this 'single integrated unit' - instead of national Armies - and - if it was well managed and coordinated, then sure, that would be a path.
But it won't happen for so many reasons.
And especially, it doesn't' need to.
Most EU citizens do not want a Federal State let alone 'more integration', most efforts for rather integration are driven by EU elite, and young ideologues who also don't really understand the nature of what is going on.
What is possibly in the cards, is an EU-like NATO, i.e. a more comprehensive EU defence organization.
But as long as NATO exists and is functional, wise Europeans may prefer to have the USA 'on board' because they are 3x more powerful when the US is involved, at least for now. And European nations right now are 'taking up the slack' and investing more heavily, Poland, Germany, Sweden etc. are really stepping up.
Some kind of Military EU coordination of forces, yes, but 'EU Army' - no. Almost nobody is going to fight for the EU flag. Also, nations want to maintain some kind of control over things like 'Intervention in Libya'.
But - we already have defence, it's called NATO, and it includes Norway, the UK and the USA.
Even with an 'EU Army of Sorts' - there is no clear leadership. Germany has the political and economic power, but little in the way of forces. France is probably the most prepared but are still small. Absent the UK which would probably be a helpful leader ... there's not much there.
In the end, NATO is well designed for that, it includes the US which is important for now.
It will take a decade at least for EU to sort itself out in that area and it may never happen. It also does not need to happen.