Okay, but this was a more complex situation than a single "give me better prices!"-persuasion-roll.
So such examples are useful to see when to change the mode of play.
So, let's go a little more complex - let's see ... in my current campaign we had a discussion about rewards with the local head of the trade guild (a Giff) - but that was solved with melon bribes. We had an interrogation trough the town guard after the PCs were found over the dead body of a victim of a space clown - but that was quickly solved with truth spells (a problem in D&D - having truth spells for formal settings like trials and interrogations).
I think I have to go back to the campaign I run before to find a social situation that could be complex enough:
"Therapy for the Fey Lord who kidnapped the Warlock to take as his bride"
So, the Party Warlock (Fey-Patron) was nearly killed like 20 sessions before and her patron came trough a portal, "rescuing" her and bringing her to the feywild (the player was absent and it was a way for her to temporarily switch to another character she wanted to try out when she came back).
So the party was looking for ways to get into the Feywild to bring her back.
So after some other adventures that involved fighting of an invasion force that had a Kraken, they finally acquired the means to go to the feywild.
And going there, they found out, that while in the game world only some month has passed, over ten years had passed in the feywild. The Warlock was now the Bride of the Feylord and Queen of the Feywild, they had a child and the marriage was like already over.
The basic gist of the coming quest was: The feywild was under attack of an otherworldly corruption and the Fey were loosing do trough poor leadership and infighting (and in actuality, betrayel! But we will come to that). So the party was supposed to go the heart of the corruption and root it out, so they can take their Warlock back home (who now feels responsible for the feywild as a the queen and living there for ten years ...)
So before they leave the next day on their quest, there is a big feast and the Party wants to find out more about what is going on, so they try to get the feylord in a good mood and make him spill his secrets.
So during the party they tried different things from performing, talking and putting drugs in his drinks (so a skill check for any different approach to influence him) - but I didn't need to roll a social initiative or use any special mechanics - I described the feast, the players described what the characters were doing at the feast (from mingling with the more common folk to daring the feylord to take drugs with them ...) - so the basic D&D things.
It ended with the feylord on the Warlock going back to the quarters, the feylord being like hammered and high - and after a little gentle pressure from his wife (and a successful persuasion check, taking into account his condition and their relationship) he breaks down crying, spilling his secret - that the corruption took his first wife and that his first wife is coming to him in his sleep needing to be rescued- but for that he needs to sacrifice his child - so he courted the Warlock in the first place only to so he can produce a child for sacrificial purposes ... (that all played out in a shadow theater kind of way - the Feylord brought his wife to the center of his power - the theater, were the past and present of the current Lord of thos region of feywild is preserved and where it can be seen) - and he was torn about it, because he liked his child and deep down knew it was wrong, but he loved his first wife so much (which is why the corruption took the face of his first wife to influence the feylord), that is why he was sabotaging the fight against the corruption and so on.
And oh boy, do we had now a super pissed Warlock
(actually the 10 years in the feywild turned her into a Druid).
But basically, it was resolved in a way, that the feylord was temporarily freed from the influence of the corruption do to the Party "therapying" him. In order to not fall for the corruption again, in a big power move, the next day he gave his free will as a gift to the Warlock, making her the defacto supreme ruler of the feywild (at least this domain of the feywild).
The mechanics I used to adjudicate all of this was basically RAW. I mean, if you look at if with hindsight, it could be loosely described as a skill challenge, but I never told the players how many successes they need or anything like that. I just described how their actions influenced the party and the mood of the feylord and the party continued from that.
I don't think social combat rules would have really improved that play experience.
All the players could shine during the party with different approaches and use their skills/abilities wisely, while the last part was more intimate between Warlock and Feylord, because of the nature of the story and their relationship, but everybody at the table loved how it all worked out. That whole encounter took like a 5 or 6 hour session to play out (including party preparations) and it was as intense as the biggest battles we had (and in that campaign we had gigantic battles with Krakens and Gods dying and the PCs using child wizard soldiers (and regretting that immediately after getting most of them killed by attacking a freaking kraken...)).
I mean, I didn't even mean for the players to find out that way (there were other ways to figure out the secret and basically If they wouldn't have bothered to find out any of the secrets, they would have been sent into a trap to be killed off), but they thought of an approach, tried it and succeeded in a way I didn't expect when I prepped all of this and I could handle it on the fly pretty well with the existing social mechanics.
I mean, maybe DM facing, I could have used a Point system or something similiar to see how far the Feylord could be influenced, but I couldn't have really accounted for that beforehand, because the approach of the party was so different from anything I could have envisioned. They used drugs, a good time and compassion to break trough the compulsion on the feylord to not only spill his secrets but actively looking for a way to do the right thing before he was compelled and seduced to do the wrong thing again.
The 5E social mechanics that exist are so light and flexible, that they allowed me to adjucate that on the fly.