ASW was the primary role of the
Invincible class, and the reason they carried Sea Harriers to deal with Bears that would have performed a dual role of reece and guidance for Soviet submarine launched missiles.
This is a dated diagram, but still worth considering. Within the area protected by the fighters you have your ASW ships and helicopters, amphibious forces or shipping being escorted, possibly mine warfare forces.Often a carrier will be generating sorties in defence of the task group and other high value assets - which for some reason many commentators seem to forget.
ASW is still a carrier role - the Pingers use dipping sonar in conjunction with towed array equipped frigates.