Having it down to a skill roll takes away a feeling of either earned consequence or achievement. You didn't do anything. You just rolled a good number. It's empty.
I understand. But you aren't doing the talking any more than you are swinging the sword. Your character is.
A combat isn't decided by a single die roll. Player choices still have a lot more impact, especially when you through a robust and complicated environment to take advantage of. Unless you wouldn't mind ever combat coming down to a single die roll.
Some dialogues have more than one roll (and some battles don't). But even so, it doesn't change the basic premise that it is your character doing the talking. To make dialogue more like combat, there could be a series of back and forth exchanges, each requiring a roll, but it still wouldn't change your feeling of just having "rolled a good number" or not.
Which is why I prefer conversations to be a dialogue tree puzzle more than a die-roll based thing. Dice should give an opportunity to skip the dialogue tree puzzle but fail just means you still have to do it. Or the die roll should give you clues as to where to take the dialogue. Not final determiner.
Or a matter of choice of circumstances. A consequence that you don't feel you earned is worthless for game experience. On tabletop, I'd let the RP happen and only bring in die rolls when I am uncertain of the reaction to the argument or when both success and failure are equally interesting. Or when the player is uncomfortable with doing the RP (shy or some other reason). I hate making dice the primary determiner of success/failure in such things save in, as I said, cases where the player is uncertain and/or uncomfortable.
This sort of use of skill rolls is just as bad as the people that think Persuasion skill is magic mind control and a natural 20 automatically turns their completely inappropriate request to be accepted.
