Surprisingly enough, the AEGIS Anti Ballistic Missile variant is in fact designed to intercept the name on the tin. That a handful of lazy journalists write "carriers are dead" articles with some regularity doesn't mean the actual Pentagon is clueless and asleep at the wheel vs it being a superficial take of a much more complex topic.

A couple years back the Navy asked congress for funding to demonstrate taking down an inbound saturation attack of 500+ missiles, probably as a deterrent to China. Congress declined.