Why one branch?

When a Canadian ship sails with a helicopter, the ship is Navy. The helicopter, its aircrew, and all the servicing people, are airforce. Their navy (RCN) handles the ship stuff and their airforce (RCAF) all the flying stuff. So a US carrier could stay in the USN, while the air detachments would be handled by the USAF. This sort of cross-branch cooperation means everyone gets along. The navy doesn't get jealous of the airforce's new toys because everyone plays together. When favors are needed, everyone has buddies in the different branches and knows each other's abilities.

How well does that work on more than one ship with one helicopter? :D

As someone who knows nothing about the military, how hard would it be to have the carriers themselves operated by the navy, and have the aircraft operated by the airforce? The airforce pilots would obviously report to the ship's captain, but the training, maintenance, and everything else about the aircraft could be done by the airforce itself.

Well the first thing I can think of is that the crew would be split (care little about the other side), and would generally be less effective at executing orders because of constant competing/infighting :)