I likewise have a dislike of determining social interactions and challenges between players and NPCs by dice rolling and have a preference towards what is sometimes called “active” or “acting roleplay”, the words and mannerisms that the player portrays of their character is used as the primary basis of the outcome of the scene.
That's exactly how we handle it as well.
However this is at least partially also because myself and almost everyone I regularly game with also do at least as much Live Roleplay as well as tabletop and tend to be pretty confident and speaking in character together. This Focus
We don't have live rp'ers in our group (well, our main GM joined a live rp I think 1-2 times, but that's years ago...) but it seems we handle the RPG pretty much as you do. For us it might be an influence that most of our players play RPG since a very long time already. So that people are uncomfortable with acting roleplay just does not happen. Even the more quiet people want to act out their characters at our table.
We do not take Cha stats into account, but when at the end of the acting roleplay scene the player rolls the dice the Cha goes into it via
the Persuasion, Deception or whatever roll. Ah, and sometimes we had a "maybe this is something your character would say, mine wouldn't".
As to players who are not so big talkers the GM sometimes takes what the player MEANT his character to be saying instead of what the player actually SAID (at least if the player said something stupid ^^). With people who are bigger talkers he is more "strict" on the other side. On the other side nearly every player had already his "ah that's what I said, but my character said that correctly" moment. And this was always accepted. It is about acting your characters out - it is NOT about getting grades for it ;-)
As to new players or people who only join for a one-timer our main GM always says "say as your character would say, don't worry about speaking, you will get into it fast". And he was always right about this.
Ah, one more thing there is - We at least try to have only one person speaking at a time. Especially for the more quiet people that helps a lot, that the "much-talkers" don't dominate. If someone currently has a scene, let him/her have it. It is also very important that the GM knows the characters well, what their intentions are, what sort of scenes their players love, so they have chances to achieve such scenes.
To people who advocate "only roll dice" I suggest - just try it once the other way. Maybe you will be surprised. And it is much more fun this way ;-) I actually don't care if people speak 1st or 3rd person, as long as they bring over what exactly their character is to achieve. If someone is like "I talk him into doing this-and-that" he needs to say with which arguments... anything else will get a "Why are you rolling dice?"
from the GM at our table ;-)