I think the saving throw idea works in a more elaborate system of social interaction, in a version of D&D that highlights tense audiences with various powers, factions, enemies, and potential allies. Specifically, it would work for if the communicated something that the target would be offended or otherwise triggered by. It would also work for things like trying to appeal to someone's emotions.
However, it fails when it comes to manipulation. Most social interaction has a facet of manipulation, but in a non-negative way. We reflect who we are talking to, try and match their energy, and try and say things that will illicit a positive response from the other person (or a negative response or something similar). Thus, if a player is trying to persuade someone by applying pressure to them or trying to hit a certain button, or is otherwise communicating with a stoic or a creature who has a different language or alternative mode of communication, a skill check is better.
The reason I think both saving throws and skill checks would work in this essentially political-drama version of D&D is because it makes it feel more like a back and forth similar to combat. This is necessary because any social drama system that's mechanically designed needs to be able to simulate the back and forth between two or more people, and this back and forth is essentially conflict, and conflict and combat might as well be synonyms when it comes to storytelling and game design.
You could also bring Intelligence, Wisdom, Strength, and Dexterity into it too in different ways. You could use Intelligence to try back up your points or Wisdom to understand someone's underlying emotional reasoning. You could use Strength for Intimidation and Dexterity for sleight of hand to embarrass someone. You could make these into saves too, where you force an Intelligence save by hitting someone with a riddle or trying to talk them into knots, or force a Wisdom save where the target's failure could cause them to lose their composure, ala L5R 5E. Strength and Dexterity might not require saves in this kind of system, but that's fine -- after all, the focus is on the drama and politics, not so much the physicallity of things (which is a secondary concern, and thus doesn't need the huge depth to it the other three stats would in this kind of system).
A lot of people will argue this is unnecessary. I don't think that argument is worth entertaining at all. After playing games like Mythic Basitonland, I've come to realize that any given rule is unnecessary to a table who plays a certain way. But I certainly can see the appeal in an in-depth political/dramatic D&D, playing out stories like Hamlet, or A Song of Fire & Ice, or even L5R. You could bring in the Honor stat, have mechanics for attacking people's reputations/honor, and give creatures stat blocks that represent their common argument tactics as actions and have bonus actions where they try and triangulate the player with another NPC or a reaction where they snap and force the party to reconsider their direction. This kind of game would be pretty fun IMO.
The issue is that all of this is basically discussing a more in-depth system, one that would be pinned on top of 5E (or any version of D&D). There are some versions of this in the DMG, and other systems have approached this topic from multiple angles, but ultimately, all of this runs counter to the Class design in the game (and only the Class design). Because most classes have almost no dead levels between 1-14, and because all non-combat features are relegated to being ribbons, you would have to use Variant Class Features to achieve this kind of idea. But that would be a very rich game, IMO!
For example, you could have the Ranger have a variant class feature based around remaining calm while in the wilds, which makes it hard to trigger the Ranger or force them into saying or doing something they don't want to do. You could have a Barbarian class feature where, instead of Unarmored Defense, they add their Strength/Con to Charisma checks to impress, intimidate, or recruit others. Spellcasters have an easier time with this because most spells can be turned into some kind of convincing display for a political/dramatic game. For example, you could cast hypnotic pattern when the audience with a dangerous faction goes wrong, or use silent image to distract someone who is debating another PC, giving them advantage. Druid's would be a blast in this kind of game because you could use their insect-shapes to be fantastic spying tools.
Overall, I think this idea is fruitful, and maybe one day I'll find the time and energy to turn this into a formalized system, complete with variant class features, geared towards more political/dramatic D&D. I could imagine this working very well with certain adventures such as Wild Beyond the Witchlight or even a modified Storm King's Thunder that focuses on the Storm Giant's political intrigue.