CWIS was developped as a solution to the threat of sea skimming cruise missiles after the USS Stark was hit by a couple of Exocet. It's not really geared towards fighting a swarm of light and slow adversaries.
A small-caliber gun with a good fire-control system or a couple of marksmen should be a better answer. Asymmetric threats have been a major concern for modern armies at least since the first Gulf War. The threat used to be swarm of small boats. Now it's also swarm of UAV. Answers exist to both.
Regarding the most likely origin of a UAV swarm targetting a US carrier close to the shore of the USA, I will go with the US army. It's good way to test both your offensive and defensive capabilities at the same time.
CIWS have a limited amount of rounds, and and are designed to stop a few large projectiles (think a cruise missile) sent sequentially. I'm not sure how they'd handle many drones, especially ones that could change course.
Throwing "wave after wave" of anything eventually overwhelms a defender.
Likely but than there's nothing left for the next wave. Cheap drones could quickly use up all the rounds. You wouldn't even have to use advanced drones with weapons. CIWS have to kill every drone just in case. Add a few missiles and or better drones in wave two and you got a very effective mix.
