The problem isn't that the fighter is so ineffective outside combat, its that the rogue and wizard ARE so effective on a consistent basis IN combat.
I don't agree with this, because it would be easy to have stuff that made the Fighter badass out of combat, but it would require new things, or going back to old stuff, like how a lot of magic items in 1/2E could ONLY be used by Fighters.
I mean, bring that forwards - make it so Gauntlets of Ogre Power, Belt of Giant Strength, they REQUIRE Fighter - maybe Vorpal Sword is just a sword +3 for anyone BUT a Fighter. Boots of Leaping only function at 50% power for non-Fighters. etc.
Maybe get some other less-advantaged classes in on that action.
Apply Remarkable Athlete to ALL Fighter subclasses, make it count as FULL Proficiency bonus (and count as trained Athletics).
Give the Fighter "Eye of the breaker" or something, where he can size a door or object up, determine it's weak point, and get advantage to hit AND to damage it, and so on.
This stuff could really add up to a decent non-combat presence WITHOUT making him "not a Fighter" (Battle Master does have one interestingly bizarre ability where he can stare at someone and know if they have higher or lower stats than him in various areas, that's another good one).
EDIT - We could have "magic item proficiency" - with different classes proficient with different items.
All of this because years of videogame conditioning has led to the expectation that all characters have to kick equal amounts of butt in a fight and the game revolves around a series of fights. It isn't rocket science. If the game is about monster fighting then all classes have to do it well which leaves fighters as last picked for the soccer team. Its the bed WOTC made when they decided killing things was the focus of the game.
Let's not pretend the elves created the gods, here. D&D came up with and perpetuated the idea that adventures should "revolve around a series of fights", particularly in later 1E and onwards (earlier versions of D&D still had a lot of fighting focus, though), whilst other RPGs moved away from that, increasingly. D&D inspired these video games, and it's just as responsible here.
As noted, what leaves them picked last (which they aren't but I concur with the underlying point) is that WotC are horribly unimaginative (in 3E, 4E, and 5E) when it comes to Fighter non-combat stuff, even though we can perfectly well come up with it (and to be fair every edition they do come up with at least one clever thing which they later forget).
It would be easy to divert, and I imagine we'll see a Fighter sub-class two, three years from now that does (but probably does something cretinous like lose it's extra attacks...).
