"I suppose it depends on how much response time is needed. I'm not an expert on any of this, and I assume the details are classified anyway. But it does sound as if the admirals are worried about this."
Even with the saboted DU ammo used in the Phalanx, I believe the Dutch Goalkeeper (which uses the same Avenger 30mm Gatling as the A-10) has a greater effective range. With the missiles both evasive and supersonic that'll greatly decrease the number destroyed by CIWS after penetration of the Standard and Sea Sparrow defense envelope.
It also has a far better radar, dual band. The CIWS is notorious for getting confused by the radar reflection off the sea surface.
Even a CIWS kill at close range (the only range it has) will leave a huge load of buckshot-like missile debris coming at you at rifle bullet velocity. Unlike WWII battelships and cruisers, modern ships are not armored to stop that.