As long as the 30-30-30-30 (30 battalions, 30 fighter squadrons and 30 surface combatants generated for coordinated operations within 30 days) goal is decades in the realization (if ever) the idea of single European integrated Army is a similar flight of fancy.
The US is only member of NATO capable of this level of response, and the reality is the appetite for doing this wealthy NATO members (as opposed to the Eastern Euros/Balts) is slowing crumbling and has been for a decade or longer.
There seems to be a slice of the EU population who thinks that current EU out-of-area operations are even mildly representative of major combat operations, or these operations (NESTOR, Mali, ATALANTA) aren't a real lift to the EU military staff.
Meanwhile, lots of low-hanging Euro defense reform fruit (like a EU military Schengen) goes untended.
NATO is the defense guarantor of Europe, period. It will be, regardless of how much "lets pretend" is played in Brussels or Strasbourg.
Further, the Euros would be well advised to put some new tires and an oil change into the 1993 Mondeo that is their defense capacity now, vice pretending its a McLaren.