The whole point is, I guess, to keep the "army" under European control - otherwise you are just duplicating NATO where the US runs it.
In the long term it's quite logical to think that as Europe gets closer politically so will the need for armed forces that can act in the interests of the EU - which may be different from those of the US & Canada in some situations
How that plays out v. NATO is an interesting situation which will depend on the future of Russia. If Russia continues as it has under Putin NATO is vital for Europe. IF Russia changed, or split up again, then NATO would be less valuable