Not anti carrier, but come on, 15 jets. So how many does it take to form an effective CAP to protect the fleet, one assumes you need spares on standby on deck to take over when fuel is low etc. Plus say a couple on maintenance... It does not leave much of those 15 jets to be effective. say two in the air on CAP, two on the deck as back up replacements, 2 on maintenance, your down to 9.
BTW just guesstimate figures.
Wasn't the Illustrious a bit of a self licking lollipop, it carried enough SHAR's to protect itself, so the reason the carrier was there was effectively to provide protection for itself?