I haven't digged too deeply into this topic, to be honest, but wasn't the gun added to the F-4 only because many pilots were lodging complaints about it? The F-4 was designed to be a missileer and was found to be challenging to use against the VPAF MiG-17s during close-range maneuvering dogfights.
Yes and no.
Yes for the USAF no for the Navy and based on late war kill ratio I side with the Navy. The gun was nice but the issues were more training.
The year was 1968 and the mission loss ratio was 1000 aircraft for 1 million sorties. The USAF concluded it was a tech issue and took the formerly podded gun from the belly of the F4 ( it had a gun pod from day one) and mounted it inside the her. They also looked into upgraded radar and missiles.
The Navy felt the issue was more systematic and less system. They pushed for changes in training for air to air against dissimilar aircraft and ground crew training. Founding the Topgun program.
End game pre 70 navy loss rate was 3.7 to 1 post was 6 to 1 the USAF kill rate remained at 2:1
Soon after the end of the War the USAF followed the Navy in there own training changes.
Throughout the war the means of air to air kills the as now was almost always missiles.