Carriers are more aware than any other surface vessel because they have aircraft on call. They can detect enemy ships long before a task group reaches an enemy fleet. Again,
read my post on aerial recon. Planes deployed from an aircraft carrier boost a task group's line of sight by 10 times, which means our fleets will be able to get the drop on enemy fleets. A CV increases the deadliness and survivability of all of our warships.
If you're fighting in the middle of the sea, it can take hours for aircraft to arrive from the shore, and by the time they arrive they're low on fuel and can't stay long. The advantages a carrier provides should be obvious on this front; it can get planes into the fight much quicker. You are underestimated the savings you can make by getting planes close to a fight. If a plane's servicing facilities are twice as close to a fight, then a flight of planes can mount an attack almost twice as often.
Planes launched from the shore can recon and hunt down ships, but they aren't good at supporting allied fleets because they're too far away to react to an ongoing battle. They have to already be on the way when a fight starts in order to get there in time to support a fleet.
E: DDs support CVs by being fast enough to keep up with the fleet carrier and small (better turning radius) and having the tools to kill submarines and aircraft, while contributing to anti-ship combat using torpedoes (which are light and just as powerful as guns). Cruisers support CVs by fending off DDs, which need to get close to fire torps; they also support landings with their big guns.
CVs kill ships, they kill subs, they kill other CVs, and they support landings, because they have a versatile array of aircraft. If well employed they should never come within range of a cruiser or DD, and indeed in history CVs were sunk by aircraft, not battleships.
The DD and cruiser are nothing without the CV, not the other way around. CV comes first, cruiser comes later (if at all).