>>20457220CIWS systems suck at downing anti ship missiles, even with half a dozen of them I'd not trust them to stop a single missile even half of the time.
Beyond that, anti ship missiles that hit near the waterline is not the same as a bomb hitting the armored flightdeck. Like all modern ships, carriers are paperskinned, they might take more punhishment due to their sheer bulk, but they burn well and a couple of hits from anti ship missiles will cause out of control fires the crew and onboard systems cannot contain. When you have 4000 zoomers onboard a ship with 4-5000 km's of wire connecting vital systems with a couple of out of control fires, you don't have a ship anymore, you have a nuclear reactor onboard a burning barge and the clock is ticking before everyone has to jump ship.