USN CV/CVNs do have to have an aviator as the CO. Usually, not always, they become XO, then CO on another deep draft ship - LHS and the like, then on to XO of a flattop. Move on to CO of another flattop. They do have to pass all the ship driver qualifications. Common sight to see mid-level naval aviators qualifying as OOD, ship's navigator and the like as they try to move up the ladder.
Also, usually for the USN, big ship command is the most common route to flag rank for an aviator. There are exceptions, of course, but it is rare.
Second the comments about Joe Kennedy and his useless son Teddy. (I do hope, however, that I'm still able to pound down the hard stuff and want to chase skirts when I'm Teddy's age however! Gotta give him that!)