"Working on" is a far cry from "tested and in force for decades", though. When push comes to shove, it's not clear that the EU member states would unanimously act in collective defense. Ultimately, while Europe may be able to collectively defend itself, it is certainly less capable of doing so without the aid of the United States.Against what threat, exactly? The big (physical, military) menace at the moment seems to be Russia, and European nations are now running almost continuous military exercises in eastern Europe to train for the possibility that any of those EU states might require protection from Russia aggression over the next few years. I don't know which banner they're operating under -- I'm guessing a lot of it is NATO -- but ultimately it's still European militaries co-ordinating to put the boots on the ground.
That said, realistically, the most effective way to protect ourselves against a belligerent Russia is probably economic anyway. Although some European states are dependent on Russia for energy supplies, that goes both ways, with Russia similarly dependent on Europe for having someone to pay for its natural resources. Neither side could trash that relationship today without suffering severely for it, but in the long run it favours Europe (because energy supplies are generally trending towards nuclear and ideally renewable sources anyway, giving Europe a credible long-term alternative, while Russia has relatively little other than energy exports to support its economy through international trade today).
Even setting Europe aside, what would you suggest for East Asia? Should Japan be allowed to rearm, potentially disrupting half a century of cordial relations and trade in the region? Should South Korea be left on their own to defend themselves from a North Korean army that outnumbers them two to one?
If you're going to make this kind of argument, you have to consider the effectiveness of armies. Bigger does not necessarily mean more effective if the little guy is better trained and/or has better technology.
More generally, I don't see why every nation should not be entitled to have a military force sufficient for its self-defence needs. The last world war ended 70 years ago. Concepts like expecting Germany or Japan of 2015 not to maintain effective militaries because of what Germany or Japan did in the 1930s and 1940s are obsolete.
In the modern world, I don't see why local agreements for mutual benefit could not provide similar assurances of defence and ultimately safer conditions than having the whole world depend on the US as you seem to want. It would take time to make the transition, but it would be better for everyone if that switch did happen over time. IMHO, that includes the US itself.