Why European carriers still need logistics

Would a European navy have to be built around new carriers?

Not entirely; carriers already exist, but useful power projection also requires a substantial auxiliary and logistics fleet.

France and Italy already operate carriers, so a European force would not begin from zero. The harder problem is sustaining distant operations. Britain's leverage comes partly from the Royal Fleet Auxiliary, whose scale gives British forces a mobility and endurance advantage that visible combat systems alone cannot provide. A carrier without replenishment, repair, and supply capacity cannot translate its flight deck into persistent power far from home.