Carrier radars are not "just for air traffic control." The CVN needs its own way of being able to see its surroundings and cue its own self-defense weapons. Technology evolves, and the means to do this evolve with it. The reason carriers are getting SPY-6 is to replace other radars that are older and have the same job: letting the ship see what is around it.

As another poster mentioned, redundancy is a thing. Suppose you don't have an E-2 up and you need to launch a fighter alert. Someone needs to direct that intercept and it's better not to have a single point of failure. Better for those fighters to have the ability to be directed from an E-2, or the CVN, or the shotgun cruiser . . . whatever makes sense at that time.

And the Navy trains for emissions control or EMCON for short. There are tactics, techniques, and procedures not appropriate for discussion here about how ships and formations of ships are expected to do their business when it doesn't make sense to be radiating sensors.