A player role-playing as a Fighter could try to aid their party by exploring or socially interacting with others during an adventure. There is nothing stopping the player from doing anything with these two adventuring pillars. However, the player is left with the question, how does the Fighter go about exploring or socially interacting with others compared to the other classes?
Perhaps by telling the DM, "My PC would like to tell the guard about X (perhaps, for example, his background as a soldier) in order to get the guard to do Y (perhaps, for example, to get the guard to let them pass without having to sign "the book" or pay the full toll or whatever). Etc. Engaging with the 5e play loop allows a player to have their character try most anything.
Level Up tries to address this question by providing class features that cover those two pillars in addition to the combat pillar. Often in a class specific way. A Fighter is going to do things for the exploration pillar that are different than how a Ranger goes about using the same pillar. They are also going to socially interact with other folk who are like them in some way, thus making them at times, the party's face in certain situations.
Sure, having some specific mechanics can also serve as a prompt to have your character do/try certain things.
My players usually don't try to be overly social unless they're playing a character with decent charisma and/or the skills for it, because they'll inevitably push for something that'll require persuasion and It would've been more "mechanically advantageous" if they had just had the face character try it instead.
Fair enough. When a PC is attempting to do something in the fiction, it is smart play to lean on their strong points in case the DM calls for an ability check which, by definition, comes with a meaningful consequence for failure.
Whereas, yeah, with A5E your Berserker might be able to use Constitution for all their persuasion checks, or the fighter might get a +d4 bonus to interacting with soldiers.
I guess I would wonder why CON can be used for a Persuasion check in the fiction? I'm sure there is a way much like one might use STR for Intimidation. Along those lines, where a barbarian or fighter might want to flex in an attempt get an NPC to back down, the DM can call for a STR(Intimidation) ability check. It's less prescriptive that way. A bonus given to the fighter when interacting with soldiers would be the Advantage mechanic in 5e... if a roll is required at all.
Yes you as the GM could throw them such a bonus when you make the call for a check, but if the player knows they have this bonus beforehand, because it's a feature they chose, they're more likely to try in the first place.
Of course. A 5e DM should not be stingy when it comes to ability checks. Be flexible with adjudication if it makes sense in the fiction - and there's plenty of ways to get different approaches to make sense in the fiction. In such a way, the player knows beforehand that said bonuses are possible depending on the scenario.
But, I get it, some people want the level of crunch spelled out for them. Nothing wrong with that, either.