A NATO E-3 force probably stands more chance of being successful as it is mainly engaged in what your man in the street would regard as "non-offensive" work.
(please hold fire electronic warriors - I'm in the realms of public perception here)
So if all we expected an EU MPA force to do was observe - there might be some mileage.
Trouble is - when we want to drop something dangerous we would probably have to have an EU summit first - and still be prepared for some of the crew to conscientiously object when the time came, or be instructed that such action was not in their National interests. - they don't all think like we do.
And as far as supporting our deterrent - I'm not sure I want to share that with some conscript from another Country.