The thing about this question is "More involved social mechanics for what purpose?" Because I can think of a number of games with very good social mechanics, starting with Monsterhearts and Smallville - but the reason those social mechanics work is that they are integrated with the setting and the social mechanics of Monsterhearts' backbiting high school cliques are not at all what would work for travelling adventurers while Smallville makes relationships and values the core of its rules which is not D&D at all and that's a very different game from D&D.
I think my ideal D&D-style social mechanics would probably be lifted from Apocalypse World with fairly forceful social interactions that fit roving violent adventurers most of the time, although the best generic ones probably belong to Fate, mostly by tying in to the character's values and abilities and where they spend fate points.