Is the US power projection at the long term benefit of the EU though? The consistent destabilization of the Middle East for instance brought more tension to Europe as people had to flee from the region. EU member states getting dragged into US wars didn't help our image in the world either.
When it comes to defense, UK and France have enough nukes for effective deterrent. The EU is also large enough to thwart aggression at its borders and buffer zones like countering the Russian invasion of Ukraine, if the EU chooses to develop and maintain a joint defense infrastructure. Unfortunately that was postponed instead of forwarded after Biden won in 2020 and EU politicians went back to the status quo.
"Power projection" isn't necessary for a strictly defensive alliance.
The fact that a direct attack on one member counts as a direct attack on all members (some of which have nukes) is enough.
Unless you're going to nuke an invading force then many of the smaller or less well equipped NATO member nations are going to rely on the power projection capabilities of the bigger militaries in the alliance, right?