Purely speculation on my part, but the F-111 would probably have still been given an ‘F’ designation even without the antichrist McNamara’s intervention. Giving aircraft that were strike/tactical bombers the ‘F’ designation under the purely nominal theory that they were ‘fighter-bombers’ when they were never going to be used as actual fighters was fairly common. The Republic F-84 Thunderjet was at least in theory supposed to be a fighter-bomber as designed, but was used almost exclusively as a tactical bomber. The Republic F-105 Thunderchief which replaced it didn’t even seriously bother to play up any ‘fighter’ angle as a ‘fighter-bomber’ in its development:
It was originally designed as a single-seat, nuclear-attack aircraft;
As a follow-on to the Mach 1 capable North American F-100 Super Sabre, the F-105 was also armed with missiles and a rotary cannon; however, its design was tailored to high-speed low-altitude penetration carrying a single nuclear weapon internally.