Notaboffin
A CV is a high value target, and to carry out offensive ops it needs, itself, to be safe. Due to the high (extortionate?) cost of such HVUs we cannot afford to get the defence wrong once - in other words our political (power projection is a political tool) tool must be almost guaranteed safe to operate before we deploy it. That requires the fleet centred on the CV to have effective AEW and CAP, tanker (force mutliplier) and ASW, after these self defence mechanisms are in place the rest of the air arm is offensive and does the power projection, probably under the umbrella of local (at least) air superioty gained by the air war part.
A surface mounted SAM platform is an inner ring defence asset, which should be taking out whatever leaks through the outer rings - relying on surface ships to totally counter an air threat is the sort of stuff that was (rightly) ridiculed in the 1920's, and took 2 decades to permeate through the ossified brains of the battleship mafia.
I am amazed that the gunnery officer appears to still be ascendant in the modern RN, after all it's some 90 years since Billy Mitchell proved the fallacy of the 'ships can defend themselves against aircraft' argument.
Dave