No agreements on any of your points. If you look at all the "Century Series" fighters in use when the US went to war in SEA, one was a fighter (F-105) in name only, one (F-104) was a short range, point defense interceptor, two (F-102, F-106) were designed to take out bomber swarms using purpose built (some even nuclear) missiles never designed to be used against fighters, and one (F-101) designed as a long range escort fighter that eventually joined the F-102/F-106 in ADC duties protecting the continental US. Only the F-100 could have been considered an air superiority fighter, and it was a bit long in the tooth by the late 60s. The first US air-to-air victory in SEA was a guns kill of a MiG-17 by an F-100.
As I'm sure you know, the Phantom was a USN aircraft foisted off onto the USAF by McNamara as part of his "commonality" concept. When USAF eventually got their unique Phantoms, they did have a 20mm cannon, but their pilots were not trained in air-to-air gunnery. I remember reading Col Robin Olds refused to allow his pilots to use guns against MiGs early on for fear their lack of training/experience would lure then into a turning fight against the lighter, more nimble MiGs.