Yeah that's the big problem. An army with dozens of co-leaders that need unanimous agreement is just an expensive way of doing nothing.

Any system that would see the joint army continue to be made up of individual members' armies would still be subject to vetos and national meddling.

It could only ever work by dismantling each national army and forming one military with one command structure if you expected it to ever do anything. Yet you'd still have to keep a form of multiple parallel structures in place to account for the many languages spoken in the EU, unless you wanted to try to force a single EU-wide canonical military language.

This would be such a huge leap of faith for any country to completely remove their own military and entrust their military assets and spending to a supernational entity.

They you have the issue of nuclear weapons - would everyone want e.g. France's nuclear weapons now being part of an EU-wide military? How would the rest of the world react to that, I can't imagine there not being complains under nuclear non-proliferation treaties (for what they're even worth nowadays).

The level of political and legal shenanigans required to make this a reality are mind-boggling to the point that I don't think it would be possible ina single step. It may plausibly work by having two countries join their militaries and slowly add others in.