I say the following as a former officer in the French army, which I'm sure you'll agree is one of the more active armies in the EU. We nominally have country-level armies. Their funding is consistently reduced and their numbers regularly downsized, and this trend exists across Europe. One might nitpick by pointing to specific budget increases, but the trend is undeniable if you look at it on any reasonable time-scale.
I can think of no European country that has an autonomous standing army, capable of deploying itself to the theater of operations, resupplying and sustaining combat efforts without the support of US logistics. To point at the mere existence of national armies is to play word games. The EU members have outsourced their defense to the United States, and are now paying the price.
EDIT: to be clear, I am quite skeptical of an EU strategic command.