What would a fighter who is versatile outside of just combat look like?
A Fighter character outside of combat is as versatile as any other character is when outside of their area of expertise. Meaning they are average. (Of course we roll Ability Scores so there is some variety within that average.)
The catch is: a huge portion of the campaign world can be assessed and appropriated for use towards combat. And that's still within the purview of the fighter class. Think like a Fighter! That's the call of the Fighter class and its subclasses.
We're at court meeting the king. Think like a fighter.
We're at the bar drinking. Think like a fighter.
We're haggling over prices. Think like a fighter.
We're worshiping at the local temple. Think like a fighter.
We're playing around with a magic item. Think like a fighter.
We're brainstorming for supplies. Think like a fighter.
We could use a different kind of weapon to exploit a monster's weakness. Think like a fighter.
And so on. The real focus of the game is on the systems supporting the classes it offers. That means most everything in the game is going to have statistics for combat (and magic, clericism, thieving, plus probably trade, crafting, etc.). Assessing those stats for combat is the game for someone playing a fighter.
But what about people who want to improve at other things for there own sake? (and we're not talking multi-classing here.) Players can try their hands at trying to be wheelwrights. Or run an inn for awhile. An easy one is to break into merchantry, at least at high level, since they likely have the funds and even goods to make a go at it.
The thing is, the players are the ones who get better at the game when engaging in areas outside of class. It took many years for the PC to become a fighter and taking a new one, especially a custom one built with the DM, takes a lot of work.