No, you haven't.
You have built a good case that the RN had more 40mm pom-pom barrels mounted than the US had 1.1in barrels mounted and you have shown that the RN may have had more even lighter AA mounted.
You have not shown that RN was actually ahead of of the US in larger DP guns, either in barrels mounted or capability of those barrels.
You have not shown that the RN was ahead in fire control, that is that their directors PERFORMED better than the US directors.
Radar is a sensor, it can feed more accurate information to a director than an optical sight/device can but if the director and or system of getting aiming information to the guns is defective or performs poorly the radar alone may not be able to make up the difference.
Many countries were working on advanced AA mountings for ships and advanced ideas for directors. The problem was turning the ideas into actual working hardware that would work and CONTINUE to work at sea. Another problem was that many of these "ideas" were tried one at a time instead of in combination or were tried too many at a time and so failed to keep working when needed.
The Germans who you seem to claim were so far behind not only had tri-axial stabilized mountings for their twin 8.8 and 10.5 cm guns but had a tri axial stabilized mounting for their twin 37mm AA guns. They paid a penalty in that the big gun mounts weighed as much as a British low angle twin 12cm mount. THE 37mm was let down by the fact that the guns were not automatic but were miniature big guns. Each round had to be hand loaded. There were also maintenance and reliability issues with both mounts, but the "idea" was advanced. The 37mm mount was also heavy and hand powered.
The Italians had a quad stabilized 90mm mounting but again it was stretching what was possible too far. And, again, sophistication has a price. Was 19 tons too much to pay for a single 3.5" AA gun no matter how sophisticated the mount? They were also working on RPC. Some Italian destroyers had both a pair of 40mm pom-pom guns and a pair of 13mm MG mounts, not either/or like too many British destroyers.
The British were certainly aware that the Japanese destroyers had main guns with high elevation, they may not have known the rate of fire or loading angle problems.
The cost associated with good AA defense was not the cost of the guns themselves, or even in some cases the mounts, but the cost of the increased size of the ship needed to carry the AA armament. For every ton of armament weight you need several tons of ship to support it and for every foot in hight above the water you need either ballast weight or a larger hull to maintain stability. Adding several tons of "improved" AA director on top of the bridge can call for extreme weight savings in other areas or a new hull in future classes. With destroyer tonnage limited by treaty Navy's had to accept limits on capabilities or limits on numbers.
The RN's .5in MG mount was an overweight piece that provided little firepower for the space and weight it took up. Awkward to use, it might have been a viable weapon in 1930 or so but by 1939 it was of little use. The tonnage may have been better used by free swinging mounts like the US was using on it's .50cals or the Germans were using on their 20mm guns even if they weren't the equal of the later Oerlikon
I don't want to continue to drag this this thread OT, but...
RN AA proved that it was effective in the Med and in the Arctic. Even over Dunkirk, the RN claimed 35 kills using their prewar AA fit including the much maligned quad .5" and the RN did not seem to make greatly inflated AA claims, as in some other navies (Off Crete, they claimed 22 and probably got close to that number and during Pedestal they claimed 66 from fighters and guns and got 42). The RN's AA FC was vastly improved by Oct 1942 over what it was in 1940 or early 1941, both in quantity in terms of number of AAA and quality, in terms of FC. The RN still had a lead over the USN in OCT 1942 in numbers of DP guns carried and in AAA, when the calibre is considered, although overall AAA numbers had equalized. The RN had advanced tachymetric directors with radar controlling most pom-pom mountings on cruisers and larger, while the same ships also had AA FC radars with gyro tachymetric directors for their DP guns. USN DP guns had been largely ineffective prior to Santa Cruz and even at Santa Cruz it seems likely that the 20mm guns were the big killers, and the 40mm bofors was still a very scarce item as was the Mk51 gyro director for bofors.
The KM did not provide director control for any AAA mountings, and neither Bismarck's nor Tirpitz had a stellar record against aircraft, with Bismarck being unable to make a single kill. Neither the KM, Italian navy, nor the IJN ever developed AA FC radar or provided their close range weapons with gyro based tachymetric AA directors.
In the Pacific, Force Z matched the USN, in numbers ofAA kills for the USN's first 3 major carrier battles, which featured massed defence by the USN's best ships, while Force Z had to make do with 3 destroyers with no AA FC and an elderly battlecruiser that barely matched a modern RN light cruiser in AA capability, and even Prince of Wales was having major equipment problems. Given equal numbers of equivalent ships it seems likely that the RN could have exceeded the USN's AA kill rate at Santa Cruz because the tachymetric radar controled pom-pom was a more effective weapon than that carried on the USN ships, save for the quad bofors which were still rare. If the quad 1.1" was effective it didn't show it at Eastern Solomons where it was available in quantity on both Enterprise and USS North Carolina which were the focus of the attack. In that action the USN claimed 28 AA kills and got 5 (including an F4F).
I agree fully with your 2nd to last paragraph and in light of the poor record of USN DP guns on destroyers, the RN decision to forgo a HA DP main armament in favour of 40 deg elevation but with the addition of a quad pom-pom, seems like the the correct decision, although it probably didn't seem so at the time, especially in light of the USN's wild over claiming in 1942.
Regarding the quad .5", I agree that it was ineffective in a relative sense, but simple logic tells me that a Ju-88 or He-111 that strayed too close to a quad .5" mount, would feel some major pain if the mounting even got on target for 1 second. It must also have deterred strafing attacks where the quad .5" would be presented with a low deflection shot, and could stay on the target more easily.
and was the smallest caliber that could be made to use a 1lb high explosive projectile and so meet the requirements of the St. Petersburg treaty that banned exploding bullets. Work was started in 1930 when such things were still considered to be important. Rate of fire was slowly improved to the point where they got 140 rpm from one barrel so a quad mount was decided upon to "equal" the rate of fire of a single .50 cal MG. Granted each hit would be much more devastating. In order to deal with dive bombers (of which the US Navy was experimenting with in 1930) the mount was given 110 degrees of elevation to compensate for roll and such when the target is coming STRAIGHT down. Because normal traverse was of little use in such situations it was also arranged so that the barrels could pivot or swing 30 degrees to either side of the dead center position without the entire mount traversing. Naturally such refinements added size and weight to the mount. This being the best being the enemy... In practice the extra traverse was either locked out or eliminated but too late to save weight. The 10 degree elevation might not have been needed in practice either. The 1.1 was a fairly powerful round for it's caliber with more recoil than the mount could stand up to initially and it had to be beefed up. While it did have tracer unlike early British 2pdr ammunition it did not have a self destruct and so didn't make those little puff balls of smoke several thousand yds out to deter attacking aircraft.
