My int is that D&D's social and exploration mechanics are not as complex as their combat ones.
I don't think that is necessarily true and it depends entirely on the game and DM.
Moreover I will say though that social interaction in particular is not always a party endeavor and often not even usually a party endeavor.
Language and situational factors often dictate which party member completes a "face check". Sometimes you can plan a face situation and strategize as to who would be best, but just as often there is no choice. Want to talk the Hill Giants into leaving Goldenfields alone, well you need someone who speaks Giant and if the only guy that can do that is your 8 Charisma Rune Knight then he is going to be the one doing it (real example from WOTC adventure).
In the example above I had a cute Rogue I built to be a face. Fey Touched background, 14 Charisma, Deception and Intimidation expertise and persuasion proficiency. Arcane Trickster with friends along with minor illusion and disguise self through race (Glasya Tiefling). Minor illusion is great to use on face checks, specifically deception or intimidation (causing a sound of whatever you are saying is waiting around the corner). It was a 3-person party with the above mentioned fighter and a Barbarian who also had a low charisma.
It was the biggest charisma check in that particular game and the one face we had could not speak their language. I think that kind of thing happens often on the social interactions.
Who gets to be the Face if there are 2 Charisma checks per session and 4 PC running a Faces?
Maybe the player who is not a face at all because he is the one who is there when it happens.
Examples from real play on the 3-face party I talked about earlier - We have 3 faces (Paladin, Rogue, Warlock), but it was the diviner who cast disguise self on himself and went into a gambling hall to get some information from the evil town mayor who was gambling. The Mayor knew who we were and what we looked like. None of the other characters could have pulled it off even though all of them are much "better" at charisma checks.
Another example from another WOTC adventure - A Gnome king sent the party to broker a treaty with some wererat Gnomes that were walled off in a corner of the city. As soon as we entered their area the wererat guards attacked us, which would have totally doomed up our mission. I was playing a mostly Bladesinger with an Undead Warlock dip and 13 charisma. I won initiative and went into form of dread (which looked like Skeletor from the he-man cartoon) and used my action to tell the wererats
"Stand Down! We didn't come to fight but if we are to fight I am happy to send your souls to be tormented by my mistress for eternity." DM-Intimidation check with advantage but no proficiency - success. There were multiple players in that party that would have been better at that check than me, but I was the only person who could try it before blood was shed.
I think this happens a LOT on social checks, if not most of the time often enough. It is why I almost always invest in charisma over constitution, even if not building a "face".
Just1 or 2 Cha checks. Not enough for 2 Faces to feel lke Faces like how you can have 2 Damage Dealers.
I think that is play style. I try to use charisma checks more than most people do at my table, to include using it in combat a fair amount and my DMs usually allow it if it makes sense.
There is a lot of opportunity if people make their own opportunity. In one party we had a Bard, a Fighter-bladesinger with a 10 charisma but with performance proficiency and proficiency in a bugle and a Barbarian with bagpipe proficiency through background. The bladesinger also had a Raven familiar with the mimicry trait.
The Bard was really the only dedicated face and the only actual good musician, but we performed quite often as a group, with the Raven making sounds to go along with the tune, my character doing the singing and occasionally the horn and using minor illusion either for visuals or sounds, the Barbarian playing the bagpipes and the Bard playing and singing and doing everything else. We did this in several taverns, we did this on a sea voyage to entertain the crew and got crew members to join in, we did it to brighten the spirits in a town after a battle. We all rolled checks for that every time. Most times we were successful, some times we weren't, but we made that opportunity ourselves, it was not part of the story and wasn't planned as a party theme ahead of time. After we started playing the game and learned each others features we decided to do this frequently. I suppose the DM could have stopped us, but why would he?