Take for instance the Harpoon, it's comparatively slow compared to the Russian stuff. But it is designed to be fired in huge numbers to overwhelm air defence systems.
Actually, all antiship missiles are designed to be fired in large numbers to overwhelm air-defense systems; that's one of the things that the US Navy's VLS systems were designed to counter. Defenses against antiship missiles are limited by their cycle times and total load -- how fast they can attack a new target and how many shots they carry. Swing-arm launchers that have to load a new missile each time they fire (or fire twice, depending on the launcher) have a slower cycle time, and may not be able to use their entire combat load in the time from detection to target (this is one of the things that drove Soviet supersonic antiship missile design, to reduce the engagement window); with a VLS system the reload cycle time is eliminated, allowing a guard ship to fire as fast as it needs to. This changes the parameters of engagement to a question of how many missiles can be carried, and produced the proposals for fleet-defense ships that were essentially nothing but massive VLS arrays, carrying several hundred launch cells each and relying on the task force's Aegis vessels to provide guidance.
While a battleship with a CIWS which has unlimited shots and can track super fast targets kind of cripples airpower and missiles (or is advertised to do so).
I can't speak for other CIWS systems, but the Phalanx system definitely does not have unlimited shots; the weapon has an internal load of ammunition (roughly 30 seconds of firing at low cyclic rate), and once the combat load is shot, the CIWS needs to be reloaded. Block 0 Phalanx would require 10 to 30 minutes to be reloaded; the Block I and later Phalanx systems can be reloaded in under five minutes. Five minutes for a Mach 2 antiship missile is 100 miles; once the incoming missile comes over the horizon, you don't have time to reload your CIWS.