Without the UK the Eurozone is a joke when it comes to power projection. I am happy to see the UK reengaging in global politics which required power projection.
I highly commend the book “To Rule The Waves” which traces the rise and decline of the RN. Notable is that as the RN retreated so did the UK's influence in world affairs. A lesson that should be noted in looking at how we have allowed the USN to decline.
I don’t think the EU wants to spend the money on power projection, as it is rarely used and the whole creation of the EU was to end their cyclical wars between them (and that part has worked).
The EU countries tend to keep small professional armies, which are only used overseas when there is a VERY clear justification (evacuating their citizens from Rwanda, for example). The public in those countries would never allow their governments to waste lives on futile nation-building to the extent we do.