Each launcher has 21 missiles. They are stinger missiles with sidewinder seaker heads. A carrier carries two launchers...and I believe they can be reloaded right on board.
So without reloading, they can fire 42 times. Maybe take out 20-25 missiles tops, probably more like 15-20. Now that's a lot of missiles but not that many in some scenarios. Wonder how long a reload takes, especially under fire, and how many reloads they carry? Still you'd have the same problem of reloading with the Phalanx, cause that drum of ammo isn't going to last all that long if there are lots of incoming.
Although I'm a little confused. What's wrong with the Stinger seeker? Sidewinder is an older design, but both have been updated multiple times.