Notwithstanding all the options in the latest US defence discussions1.....
The Kennedy has been chosen for the downgrade from 12 carriers to 11 primarily because it's been slated for a complex overhaul (COH) since she returned from the Med in December 2004. It's going to be an expensive proposition for anyone, NATO or otherwise, to keep her operational.
But the most serious limitation is NATO itself. I'm sorry folks, but having worked in NATO HQ, it's a tragic sitcom just waiting to happen. The Navy is the military element least amenable to flexibility within a chain of command (STANFORLANT notwithstanding). The Captain's orders would have to be vetted by a committee, and even though everyone speaks English, the order would have to be given in French. Merely turning into wind for air ops would require standing up a study group.
Not going to happen.
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1. Google - The Conference Report On H.r. 5122, John Warner National Defense Authorization Act For Fiscal Year 2007 -- (House of Representatives - September 29, 2006)