In an old Warship magazine volume, there is a good explanation of the development of the 5in/38 DP gun

The 5in/25 AA gun was developed so that it would have a low inertia to training and elevating, as well as a low muzzle velocity so that shells and brass cartridge cases could be handled as a single piece. The weight needed to be below 100 lbs so that they could be manhandled. The 5in/25 had a power rammer, so the gun had a rate of fire something like 25 rounds per minute. The 5in/51 gun, which was used in the secondary armament of battleships, used bagged charges, so the rate of fire was something like 9 rounds per minute. Heavy cruisers were armed with the 5in/25 gun as their secondary armament, so they were reduced to using it in the anti-destroyer role. The problem was, with a muzzle velocity of about 2,200 ft/sec, the danger space was too small. The answer was to develop the 5in/38 gun with a higher muzzle velocity, but still short enough to fit a dual-purpose role. The successor to the 5in/38, the 5in/54 gun, as was used as the armament on the Midway class carriers, was a balky gun. The Montana class battleships would have had the 5in/54 gun, as well. They were the originally intended recipients. The 5in/54 had reliability issues, and used heavier ammunition. The 5in/54 gun used later is a considerably different piece.