The Japanese could not actually build what they wanted, they could not afford it.
They had gone to the 5.5 in guns because the lighter weight shells were easier to manhandle to the guns from the hoists.
Land-based artillery in the modern era has pretty much standardized on the 6" size, as that's about the largest shell that can be manhandled if necessary, and also that's about the maximum size of a gun that can be installed on a SP platform, provide some splinter protection for the crew, and still be road mobile.
For sea-based artillery conditions are more cramped, and the platform is moving and pitching, moving the maximum size that can be manhandled downwards. Consensus seems to be that somewhere around 5" is the maximum that can be manhandled in a somewhat sustained fashion without tiring the gun crews too fast.
Now please note that this was a limit to the British 5.25in AA guns which failed to reach the desired rate of fire.
Yes, the Brits waffled around with a number of different guns in this class until they developed and then standardized on the 4.5", which by all accounts was an excellent gun.
For all of their emphasis on airpower Japanese Cruiser AA was pretty pathetic. It was also a bit wasteful as instead of accepting the loss of torpedo armament (or even the spare torpedoes) and just using the best AA guns they had they kept trying new guns but only mounting 4 on the Agano's didn't do much.
The Agano's were going in too many different directions at the same time.
If you have perfect air superiority you don't need AA!
Was fire control the big thing that was missing for Japanese AA, or was it just they had relatively few guns compared to the USN? They had the 127mm Type 89 which perhaps wasn't up to par with the US 5"/38, but it seems to have been Ok. And later they introduced the 100mm AA gun, which seems to have been excellent, although it was so late it wasn't introduced in particularly large numbers before the US juggernaut steamrolled them.
For bigger ships, the insistence of a separate surface secondary battery and a separate heavy AA instead of DP mounts was arguably, in retrospect, a mistake.
Perhaps the larger gap was in light and medium AA. Some of their ships had pom-poms, which weren't as good as the 40mm Bofors, in newer designs being replaced with the POS 25mm AA gun.