Some thoughts that intersect with 5e's default Social Interaction (Pictionary/Wheel of Fortune esque) mechanics:
(1) Torchbearer has a social hierarchy built into its system called Precedence. This is a scheme 0-7 of relative values for social strata. Your relative value tells you what sort of social conflicts you're able to engage with socially vs other parties.
Something like this could be ported into 5e with Adventurers getting +1 Precedence per Tier (starting at 0). Feats could either (a) improve your Precedence level (along with giving you a + Charisma) or (b) let you ignore Precedence differences with a particular subset on the Precedence hierarchy.
The difference between an Adventurer's Precedence and the NPC's Precedence = +5 DC/-5 DC per difference (or 0 difference if same Precedence).
EDIT - This can also be integrated into Background quite well: you get +1 Precedence with those of the social strata related to your Background.
(2) Conversation occurs organically in the social conflict. However, players have 3 turns at DC 15 (with Precedence differential above) in the course of the conflict to do the following:
* Reveal NPC dramatic need/leverage. This MUST be accomplished in order to move to (3). Without uncovering this, you cannot successfully resolve the social conflict.
* Uncover IBTF. Success on this will grant you Advantage in (3). A failure on this equals some complication to the challenge. The fiction changes and the situation changes. Mechanically, things change in concert with this. Perhaps a 2nd NPC shows up and actively works to end the social conflict early; the player loses 1 of their 3 turns. Perhaps the NPC is opaque and impossible to read. Instead of Advantage on (3), you get Disadvantage because they will surely take the worst reading of whatever you say. Perhaps you've attained leverage but now the NPC's partner arrives on scene and they reveal themselves to be "the pants-wearer." Now you have to learn their dramatic need/leverage in order to procced to (3).
* Subvert Precedence differential. Success is effectively flattening the social order (Precedence) by 1 for this conflict. You've said something or revealed something that knocks the NPC down a peg. Failure here likely means they take offense and their Precedence increases by 1 for any subsequent moves or for (3) (meaning the DC increases by 5).
Regardless, whatever you do, your say what you're doing and then you make your move. For instance, Uncover IBTF might be something like "I notice you've got a stable of the finest race horses. Do you ride yourself or is it beholding the beauty of the horses in full gait that is your passion." <Player then says something like I want to see how deeply she cares about these horses. If this is her great passion in life, that is how I'll get her to help us - looking for leverage> Player rolls Wis Insight DC 15 (NPC has +0 Precedence advantage).
The player has spent 1/3 turns. If successful, they have leverage and they can move to (3). If not, they're going to have to spend further turns on getting leverage. Further, the player can immediately go to (3) once they have leverage if they don't want to risk failure on any of turns 2 and 3 and complicating their situation.
(3) Straight-forward. You've got leverage. Now you make the Charisma (whatever) check to cement the social conflict. Say what you want (using the leverage/dramatic need and any IBTF that you have successfully uncovered) and roll your dice. DC 15 +/- 5 or more for Precedence difference with or without Advantage.
Success equals you get what you want.
Failure equals the NPC either takes offense and now you've got a problem (perhaps you lose 1 Precedence for those in this NPC's strata until you do something to resolve that...perhaps future random encounter tables are now populated by this NPCs cohort...or perhaps they show up as complications in other related social conflicts) or they give you what you want but they want something significant and immediate in return.