If the role of the embarked aircraft was purely maritime, for example air defence of the fleet (SHar), anti surface / sub surface (helos) etc then it makes sense for it to be FAA as it is a purely maritime effect.
In other words, keep the RN at Sea and forbid it any influence over events on land. By this logic, shouldn't the Air Force limit itself purely to air to air engagements, ie QRA/no fly policing, since anything else effects the Land Theatre, and that's Army!
I think that this is something that the RN doesn't really understand - the RAF isn't trying to destroy the FAA, it simply believes that it should be responsible for providing air power.
It could also be argued that the RAF fails to understand that neatly separating the battlespace into Land, Sea and Air and allocating them purely to the Army, Air Force and Navy respectively doesn't work - there are always overlaps. Amphibious Ops, Maritime Patrol, CAS, Carrier Strike and even Base Defence (RAF Rgt?) all fall into these categories - operations from one environment influencing/supporting operations in another.
It would appear that the concept of "Jointery" isn't as advanced as we would like to believe.