But you're right, just leave that darn Navy ship to do Navy things as opposed to providing a persistent, high fidelity, AD capability while we run expensive, non-persistent CAP from Cyprus. Assuming they let us do combat ops from there.
The clue's in the name -
Sovereign Base Area. We can do what we like from Akrotiri, no need to go crawling to the Cypriot government.
I agree with your points about sea-based air defence being useful when most of the battle is fought close to the coast, though.