There are many approaches that don't introduce dump stats and handle social interactions well. There are tens, if not hundreds, RPG systems with no such issues. Problems with charisma are endemic to D&D, from 3e up. Thus, any sensible solutions require getting away from the faulty framework of modern D&D.
The most obvious aspect of the problem is the redundancy between skills and stats. Solution? Get rid of either. Early D&D editions worked well with no skills; Fate works well with no attributes.
If you remove stats you also free yourself from "must have" stats for various classes, stat-modifying spells and equipment and similar things. You lose encumbrance calculations, but that's not really a problem.
If you remove skills, characters can be proficient in various background activities without handwaving them or wasting resources on something that's rarely useful in play.
There is also another source of issues - less obvious, but often more problematic. It's the unclear correspondence between the content of the character sheet and the fictional character it represents. What is abstract and what is concrete? What do the stats really represent? What does a class mean?
If charisma is only a number, with no in-game meaning, then what is wrong with dumping it? If it is personality and looks, then 7 cha character will be an ugly, unsympathetic coward no matter how high his diplomacy. Don't expect others to like him, even if they agree with his argumentation.
Define clearly what the numbers mean and a lot of problems disappear.
Finally, if you want to have redundant character statistics that don't make one another unnecessary, give them other, universally useful, powers. For example, add a special ability to every point of positive attribute modifier. If charisma +1 (12) lets you reroll a failed save against fear or compulsion once per encounter and +2 (14) creates an opportunity for a good trade once per adventure (eg. buying a magical item that wouldn't otherwise be available), dumping it is no longer an obviously good solution.