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The EU Battlegroups are not the combined militaries of the member states. They are a specific group that works directly under the EU. The vast majority of the militaries of the EU member states are not in EU Battlegroups.
Did you not actually link of of your sources because you were concerned someone might actually read them?
Why do you equate “not in an EU Battlegroup” to “not able to go to war immediately”? Every EU member retains independent control of their militaries. They do not need to use the EU Battlegroup command structure. In fact they never have, and yet they have still regularly deployed troops.