Except you would be wrong. The Air Force wanted an aircraft that could serve as a fighter/bomber, and the Navy wanted a fighter/interceptor to defend Carrier task forces against Russian bomber/cruise missile attack swarms. McNamara made them combine their requirements into the "Tactical Fighter Experimental" (TFX) program, and it was a "fighter" in that it was intended to use missiles at range to shoot down enemy aircraft, in an era where dogfighting was considered to be obsolete.
No. The Air Force's primary interest was that of a bomber, and the Navy that of an interceptor. Fighter was a very secondary thing. Much like the F-117 can mount sidewinders for air-to-air, or an A-10 for that matter too. Its more a self defense thing. That does not make an aircraft a "fighter". Note the distinction between "fighter" and "interceptor", the later has more to do with taking out big slowly maneuvering bombers. The F-111 was actually envisioned as an answer to both sides of this traditional "interceptor" mission. For the Navy it attacked the bombers. For the Air Force it replaced the big low maneuverability bombers at altitude with small fast terrain hugging bombers, aircraft far more likely to find gaps in Soviet air defenses.
The F-111 was never intended to be a front line fighter like the F-4. And it was the F-4 in that era that was intended to be a front line fighter that shoots down enemy fighters at range with missiles. Early F-4 versions did not include guns.
Meanwhile, the Air Force would use it mostly as a tactical bomber, and also made an adapted version to use as a strategic bomber, and it did a pretty decent job of those - but it could have done much better had it been designed to do just that rather than trying to be everything to everyone.
That was the role the Air Force had always been working towards. And not everything, not a "fighter", just a bomber and and "interceptor". Again an "interceptor" is not a "fighter", different prey so to speak, primarily targeting bombers and such.
And I can only imagine what would have happened had the Marines demanded it be SVTOL capable, too.
To be clear I am not in favor of multi-mission aircraft. I prefer the 1970s renegade sort of approach that led to the single purpose designs of the F-16 and A-10. That is not to say there isn't more opportunity for common components. We probably could have more standardization and commonality with respect to avionics, maybe more aircraft sharing engines.
STOVL would probably have to be its own single purpose design, its hard to imagine vertical landing being a general purpose win over ruggedness. Or maybe it would be a better idea to adapt amphibious assault ships for STOL (no vertical). No need for the complex arresting gear of a carrier that supports many aircraft types. Maybe something simpler for a single type of aircraft, whatever STOL the Marines are operating. I understand OV-10 have been launched and recovered by assault ships but something closer to an A-10 would be needing more runway so arresting gear of some sort would be necessary.