There appears to be a bit of a misconception about "capital ships" and "self-licking lollipops".
Air defence, wherever it is based, is an enabler and RAF types tend to get a little dismissive of Air defence in the RN because before JFH the carriers were seen as very much a self licking lollipop (what is the carrier for - to carry harriers; what are the harriers for - to protect the carrier). Whilst this is a massive simplification there is a large element of truth to it - the RN really had no way of projecting power at range.
The statement above illustrates the problem. LPH/LPD/LSD, MCMV, all sorts of different vessels are part of a Naval force that may be mission essential units, either delivering troops ashore or providing other enabling effects. The FAA f/w force was about defending that whole and allowing it to do its mission, rather than just defending the CVS. Doing AD doesn't necessarily mean having X cabs tied to that mission for the duration either. Just as Typhoon did on Ellamy, you can go from AD to strike and back depending on the phase of the operations and the tempo. It's another reason why very good multi-role platforms (eg F14, F18 and yes, SHAR) tend to be preferred at sea.