CIWS are manifestly inferior to both existing options and basically redundant given the existence of shields.
20th century CIWS, maybe. 24th century equivalent, maybe not. And besides, we KNOW that shields never collapse or lose power.
There's nothing in canon to suggest that pulse phasers can be multi-directional and the fact that the Defiant's dorsal and ventral phasers (which are multi-directional) aren't pulse phasers somewhat argues against that.
Then mount them on turrets.
The Galaxy class was already fitted for Danube-class runabouts, so less capable impulse-only craft that would therefore have vastly inferior shielding seems counter-intuitive.
Why would their shields be inferior? I never said these ships wouldn't have a powerplant, they just wouldn't have the very bulky apparatus needed to form a warp field. You replace that space with weapons and shield generators, even a runabout sized ship could be a force to be reckoned with... it just can't go to warp.
Jellico was a rigid, pompous control freak who would order most of the improvements ripped out as "unreliable" or run the crew the crew into the ground before the first engagement "fixing" imaginary flaws.
Then you get a person with Jellico's "kick butt take names" mentality who's been trained on (and trusts) the new design.
As noted above, essentially pointless, especially by the TNG era, given the existence of shields (arguably the earlier ball turrets are CIWS or at least a hybrid of CIWS and autocannon).
In a vessel built for combat, a tertiary defense system makes sense, even if it's not quite as effective as shields or reliable as armor. The torpedo that never reaches your shields never drains them.
I've already touched on the suicidal/criminally negligent aspects of that idea above, so I'll just reiterate that the runabout is about the largest auxiliary craft which could "parasite" on the Galaxy-class or similar.
A Galaxy class also carries an arboretum, a school, a lounge, gobs of science labs, and crew quarters like luxury suites. And, reportedly, a dolphin tank. Eliminate some of that, and you can enlarge the shuttlebays.