The French have been doing policing in their former colonies several times. But for true global power projection one needs a fleet that has 10+ carriers with the airpower, amphibious and support elements that goes along with such big ships. No European power is capable of doing that plus they have priorities that USA does not have like needing "brown water" ships for local defense.
Also having a bunch of smaller nations all wanting to have a well-rounded defense might also not be the most cost effective way. But if a nation does not want to be completely dependent of other nations for basic stuff, then that problem is not going away.
But changes are happening e.g. the Dutch and Danish navies are phasing out smaller ships and replacing them with larger and more capable ships. The Danish Navy no longer have any subs, missile boats or corvettes: the good old Cold War scenario of Danish and West German navies fighting in the western Baltic and local Danish waters no longer makes sense.
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