Maybe, but I think it more comes down to how much do you want to lift them, and would doing it encroach on other classes?
IMO Fighters should be the best at Fighting. One feature I gave fighters at level 6 is Consistent Attack, which allows them to make a weapon attack with their bonus action even when taking other Actions (like Dodge or Cast a Spell). Why? Because a good fighter should always be able to get an attack in, regardless of what else they might be doing.
Now, I realize you've been talking about improvements outside of combat. But if Fighters are the best at fighting, what should they be great at outside of their bailiwick?