For all of the bickering about NATO, it might be wise to consider that we did, at one time, share a common cause and at least a common purpose whose momentum carried over into the difficult peace after a horrible war. That common purpose brought a measure of peace and stability to central and western Europe.
If the European Defense Force is seen (politically) as a replacement for the collective security structure maintained by the North Atlantic alliance, will we have to engage with our children -- on both sides of the ocean -- on the pros and cons of that linkage and the value of being on side together?
This issue came up 20 years ago as "Eurocorps" was being tossed about as an idea. As an RRF, the idea was similar to UN and NATO in terms of nations offering troop units to support an op. Someone mentioned the NATO AWACS group, who have been an operational success. Another multinational success was the WEU maritime interdiction by in the Adriatic (Sharp Fence). It was later merged with a NATO op (Maritime Guard) to become Sharp Guard which was a success story insofar as multinational operations go.
Can this multinational force operate without the core C2 backbone from NATO?
As the nations have gotten used to working together for about 70 years, maybe yes. .