things have gone very badly wrong if a carrier is firing anti air missiles
You reckon things always go as planned in War?
Think Hood, Singapore, daylight bombing, attacking with armor up a single roadway over multiple bridges, and so many other tragedies!
Going into a fight with just your Willy in hand.....is not a smart thing.
Arm them up....as you cannot afford to lose even one of them.....as your carrier fleet takes a 50% Loss.