Usual Phalanx engagement time (duration of burst) is somewhere around 2 sec, i.e. 150 rounds. Every machine gun has certain dispersion, it is design parameter. For example German MG-42 was considered better then Bren LMG because it had somewhat greater dispersion. Of course, it must not be to great either. For the Phalanx, it is calculated to form a "net" at usual distances with maximum hit probability . Btw, even one heavy round from Phalanx should be enough to make AShM un-aerodynamic and force it to crash.
Certainly every gun has a dispersion. The CIWS gun, however, is the M61 that was originally designed for fighter aircraft with an emphasis on air-air engagement. As such, it already has a low dispersion by design, for point accuracy (a fighter sized aircraft at 2.000-6.000ft distance) was required. That is it's weakness in air-ground engagements, it's so accurate that it effects only a very small area on the ground.
The latest Block 1B upgrade, among other things, sees use of a refined gun barrel with even tighter dispersion to provide greater first hit distance.
Not a shorter range per se, but less prediction uncertainty where would our interceptor (CIWS round or a missile like RAM) would have to go . In that sense, laser would be ideal because it travels at the speed of light so you could just point it at the incoming AShM . But lasers have other properties that make them unsuitable for the task . So far rail guns show greatest potential for this kind of application .
The point here is that a gun CIWS, at the time of the projetile leaving the barrel, will need to know exactly were the intercept will occur, because the bullets cannot update/manceuver in flight.
A guided missile does not need to know, it does update the calculation in flight. As such it simply doesn't matter if there's more time between launch and intercept. A higer prediction uncertainty doesn't matter then, because it is lowered in flight by the updating calculations of the missile.
Imagine AShM making 90 degrees turn at supersonic speed (like Klub AShM does) . With simple proportional navigation IR missile would suddenly need to make even harder and sharper turn , loosing speed and precious time . Klub could make maneuvers up to 10 g , and that would force RAM to go up to 30-50 g in order to stay on intercept course . I don't know what is max g of RAM, but even if it could execute such maneuver it would lack speed and time to catch up with Klub again
The problem of intercepting missiles having to make much tighter turns than their targets has been existing for a long time now, and it is the same for every missile there is, no matter what it's guidance mode. Older Sidewinders are said to be capable of 20-30g I think.
I also believe people overestimate the size of the maneuver such AShMs make. After a 90° turn, it points away from the targeted ship and will then need to make 100+° turn to come back. And there's the problem of the seeker loosing lock.
If RAM is capable of executing such a maneuver, which naturally is a design requirement, it'll maneuver to the intercept point and not fall in a chase. As such, it will never fall in the position of having to catch up. Especially since in the endgame the AShM will have to go towards the ship. Othersise all the maneuvering is useless.
Additionally, modern seekers are so sensitive about small changes in relative movement, they're hardly lacking compared to a radar tracking the target.