So long as ships are operating in internationally recognised water ways with freedom of navigation why would we ever stop them?
The English channel has long been a shipping route used by multiple nations. It doesn't upset people, that Russians, or the Chinese use it. Also, Western Europe has been dominated, by European powers. Its not like Europe is has a whole lot of examples of a non-European (non local) navy, turning up, blasting the entire continent, defeating everyone and locking them to servitude. Europeans probably can't even probably imagine that, European defence tends to dream as far as Russia, which has been a long time Europe actor for over a thousand years. Russia as a naval power has be bit of patchy history, so the fear about Russia was less about ships, but more about a land invasion.
If you really want to create a split in NATO about freedom of Navigation, ask the Canadians how they would feel about Chinese carrier groups exercising in the North West passage. Heck they don't even like the Americans going through there.
A Chinese carrier probably doesn't worry Europeans as much as say 24 Chinese submarines operating in the English channel. A carrier operating in a choke point aggressively is rather silly. Its gifting a target. Sending two dozen submarines to sight see European coasts would be pretty odd.
24 submarines operating in and around a choke point would be way more threatening. Europe isn't worried about a carrier turning up and controlling airspace. Europe would totally be worried about a fleet of subs turning up and shutting down all port activity. Sending so many submarines that you could literally walk from France to England on top of Chinese submarines would be very novel for European powers. Of course they wouldn't do that, but they could sure control strategic choke point globally if they Americans were busy elsewhere. Namely the med, the Gulf and the straits.
Chinese operating Carrier groups in the Mediterranean would be a bit different. A Chinese carrier would provide significant enough air power to be concerning to countries in North Africa and the middle east. Not enough to attack stronger nations in that region, but enough that if they were to side with an enemy in the region, it could shift the balance of power.
China wants to make others feel less sure about freedom of navigation. It now has enough naval power and skills, it can go around and do that. Just like the Americans go around the world to prove the opposite.
However it is getting harder and harder to do that against China. China can offer more and more capability to counter it. Maybe China will start fishing in European waters and send its coast guard and navy to defend its fishing fleet. Oceanic rights, EEZ, freedom of nav are all tied in together and China will happily test anyone's belief. I would imagine the Europeans would be quite upset if the Chinese turned up with oil drilling rigs and started to drill and ship north sea gas and oil without permission.