... I tend to see the EU as a possible counterweight/alternative/stabilizer to China (economically), Russia (military) and the US (both)...
Interesting observation.
If you've come to that conclusion, it is not inconceivable that the troika could have done the same. It can be argued that the biggest threat to U.S. dominance is not China or Russia, but the E.U., at least economically. It has larger population, bigger total GDP (I think), etc. What it lacks is a (truly) unified market, and free(er) movements of goods and people. Common language would help too. But it has been moving in that direction, slowly but steadily, toward a United States of Europe. If and when that would happen, it would be a serious competitor to the U.S.
Euro in itself poses a threat to the dollar's dominance as the world's single currency. First central banks around the world started diversifying their reserves by acquiring Euro, in addition to dollars. Then some countries started floating the idea to denominate their oil sales in Euros. Some would argue that as soon as Saddam Hussein publicized that idea, and Russia and Venezuela initially responded favorably, it was the end of him.
For those who like to indulge in conspiracy theories, the following wouldn't be totally improbable: if E.U. is seen as a serious competitor to the U.S., it surely must have crossed the minds of Americans that it makes sense to slow down that advance. The first attempt to destabilize Europe was to support Islamic states within its borders (Bosnia and Kosovo). Having a war in the middle of Europe was also designed to scare potential investors and re-establish the U.S. as the only true safe haven, the last bastion. Then someone must have realized that the regime-change policy, adopted by several administrations, Democratic and Republican alike, and the subsequent destabilization of many states in the Middle East and North Africa, resulted in an unprecedented surge of already existing refugee attempts to reach Europe. They couldn't find a better way to destabilize Europe. Apparently, it's been working.