While the Phalanx CIWS is inarguably impressive in specification and testing, the thing about the US military is that, whenever we get into a fight, we're always 150% ready - for the last war. It takes us a while to figure out what we're doing in the current one, and it costs us. I see no reason to imagine this pattern shouldn't hold in a new war against an opponent equipped with Moskits.
Think of the problem:
- At Mach 3, the missile is flying at roughly the same speed as the unguided bullets the thing is firing. If it's additionally doing evasive maneuvers it's going to be pretty hard to hit.. (though I guess at Mach 3 it's not going to do very much acrobatics, but perhaps even a little bit is enough to make the CIWS miss?)
- Mach 3 is close to a km per second, and I guess the realistic range for a 20 mm cannon isn't that much, lets say 4 km for the sake of argument. And below, say, 1 km it probably doesn't matter if you keep firing at it or not, it's likely to hit you out of sheer inertia even if you hit it. So that gives you around 3s time to fire and adjust the aim. Not very much..
I have seen some mentions of a move toward missile-based defense, which would probably be a more effective option, but most of the same considerations apply - each missile might be much more likely than an individual CIWS round to score a kill, but many fewer missiles can be carried, and the launchers still need time to reload once they exhaust their magazines.
This is also, incidentally, the real reason why DEWs and hypervelocity kinetic weapons - lasers and railguns - are such a matter of interest in the naval research establishment at the moment. They're justified as being a more cost-effective means of dealing with low-level threats, but this is America - the last time our military services had reasonably to worry about being short of funds was no more recent than the mid-1930s, and nobody since has given any meaningful fraction of a damn about the cost per shot of naval weapons - and frugality would militate against the very considerable R&D cost of new systems without new capabilities, in any case.
A much more plausible reason why these new systems receive such interest is because they might eventually offer a solution to the currently insoluble problem of how to defend against a mass attack with supersonic AShMs. You don't see that discussed much in public, because no one with a stake wants to suggest that our enormously expensive carriers no longer offer the power and prestige that merits the cost of keeping them active, but nothing else makes as much sense.
WW2 battleships proved their worth as late as the 1990s when nothing else could deliver the shore bombardment capability.