I would say your Face player was absolutely right to be worried, because D&D is a RNG-heavy game, and if your system involved significant rolling, he was not going to do well.
All the games I can think of that do "social combat" successfully fit one of two models:
1) Dice pool or similar RNG-killers - All the WoD games for example. When you're using large dice pools, RNG is generally much less of an issue. A character invested in their role is almost always going to roll decently.
2) PtbA/BitD/Resistance-type mechanics where it's more about "success at a cost" and where there may well be meta-resources to override RNG, and where the consequences of success/failure are better-defined and less binary than D&D.
D&D 5E uses a high RNG-system without a net (i.e. no Take 10 or Take 20), which means that, very often, heavily-invested Faces (max CHA they can have, as proficient as they possibly can be in social skills) still absolutely constantly miserably fail their rolls. You roll a 2, and doesn't matter that you have a +9, you didn't make that 15 DC. The Barbarian who dumpstatted CHA and has no social skills rolls a 16 and he succeeds it. Only Eloquence Bards level 3 and above, and Rogue Faces level 11 and above don't face this issue, because of Reliable Talent.
No edition of D&D has done great here. D&D is just too RNG-oriented.
If you wanted to design a system that worked better, I'd suggest starting by adding "Passive Persuasion" (and Deception/Intimidation), i.e. modifier + 10, and using this as the basis of your system, and if any rolls were being made, making it easy for more-invested characters to get access to re-rolls, Advantage, and so on. You also need to decide what to do with Help, because if it's allowed all the time, it's easy to have perma-Advantage on social stuff, which is absolutely fine, but is something to consider.