There're definitely two different traditions for carrier leadership in the world:
1. An American one, where CO/XO and most others top positions on carriers occupied by aviators or NFOs. AFAIK, senior SWO on USN carrier is CHENG, who can be nuke LDO though. Some USN flag rank SWO, ADM James Stavridis for example, had advocated once the carrier XO billet for SWO, explaining that there isn't proper leadership on a carrier for young SWOs who are on their first or second sea tour, as 1310/1320 people can adequatelly lead only the same aviators/NFOs, and a career of young SWO can be completely broken during his/her carrier tour. But it fell on deaf ears, as far as I know.
This way is the one the Chinese PLAN follows only, supposedly. They too are preparing the PLAN aviators for the carrier command. For some reason, this tradition can be seen as the way to lead the navy with solid carrier force.
2. The British one, where roughly one quarter of carrier COs were aviators and XO is always surface guy. The most nowadays carrier fleets follow this way - France, Russia, Brasil, India, Italy, Spain etc. The top adviser for aviation matters aboard, given the often situation when both CO and XO are surface warfare officers (or even former submariners), is so-called "Wing", a Commander (Air), the leader for all aviation activity of a carrier as Aviation Dept Head. He is aviator, inevitably, and all aviation people in ship's company (lead by LtCrd (Air) or "Little F"), as well as all embarked squadrons' personnell (there is no XO position in Royal Navy Fleet Air Arm squadron - Senior Pilot and Senior Observer are sharing this responsibility) reports to him. In a sense, he is old USN CAG incarnation, when (prior 1983) CAG did report to carrier CO. "Wing" can be Royal Marine officer, a flyer too.
The main problem here is that carrier is regarded as surface warship where the embarked aviation is just a kind of ship's weapon. So it's often easier for CO to say that "this is impossible, so it's over, dismissed" than to try to take the aviation boots on and do what the situation may demand beyond the rules. This way is, evidently, unacceptable for any big carrier forces (i.e. more than one or two carriers within national navy).
The core philosophical difference is that USN carrier fleet is a part of specifical Naval Air Force, the one that has all that any air force has plus something more. This American Naval Air Force is unique but in essence this is a kind of air force with all leadership positions truely related to aviation world, no matter which kind of uniform or ranks is in use.