They are the undisputed king of attacks getting 3 and 4 when other martials are capped at 2 and the classes you mention raising/possessing the king (Wizard, Cleric) only get one attack.And if the fighter was the undisputed king of combat, that would make it OK to not have anything else unique to contribute. OD&D got that right by giving the fighter more hit points, higher attack bonuses, and the ability to use any weapon or armor in a game where acquiring random magic items was the key factor of character growth. Since modern D&D eschews that model, the fighter needs to diversify, or at least classes that support the "skilled warrior" trope be added.
If you give the fighter more you should take away extra attack. I would be ok with having an optional rule to get 2 more skills and a tool, along with expertise in 2 skills and a second fighting style in exchange for limiting them to 1 attack a turn. They could add one more expertise at level 11 and a 3rd fighting style level 20. They would be a lot more versatile then, but my guess is those who want a more versatile fighter want to have extra attack too.
OD&D was awful and far less balanced then 5E with Fighters being WAY to OP.