You both are somehow thinking that having a skill system prevents roleplaying which is totally false.
No, it's not, because....
It allows people who do not want to rp to also play the game by using dice rolls without any effort to play the scene out...
So it's not 'totally false' then. It is apparently not a bug but a feature?
, but in a rules less game like you propose they simply would not play at all.
That's not a given.
Likewise, the RP players are still able to talk in character all the time.
That's not a given either.
The only limitation is that to some extend what they say should match the die roll.
What??? So I'm now letting the dice dictate what I say? That doesn't really sound like freedom to play. The way I integrate free form role play with mechanical resolution is exactly the opposite of that. I let the player determine what they say, and the content of their ideas, and to a limited extent the presentation of those ideas - deceptive, appealing, threatening, etc. Then I let the dice decide how the NPC responds to that content.
But there is something HUGE that is missing from your analysis and an area you are seemingly completely blind to.
The skill system does nothing at all for the most critical aspect of really good role play - intraparty play. If you don't have good intraparty play, you really don't have good RP at all. So your guy who is now playing because solely because he doesn't have to RP because he's got lovely dice resolution is still not really integrated in to the group, and is still missing basic skills that the mechanical resolution system is not going to be able to replace.
Moreover, your entire argument is predicated on the idea that the DM is an SOB and just wants to screw you over and is entirely biased and unpredictable and apparently also illogical. If that is the case, then your mechanical system is unlikely to help, since its the DM that sets the opportunities, the stakes, and the difficulties in most systems. You are still going to be reliant on the DM's judgment 95% of the time.
Moreover, your entire argument seems to be predicted on the idea that the reason your role play is to get NPCs to do what you want them to do. The reason you role play is that this is a role-playing game. That you get the NPCs to behave in a particular way some of the time is sorta irrelevant. Role play is the primary window into the world of the shared imagination, and particularly on to the most important things within that world - the NPCs. If all RP did was let you tell who the players on the stage where, and what their role was, it would be sufficiently worthy in and of itself. "See I smile and smile, but am a villain.", "a rascal; an eater of broken meats; a base, proud, shallow, beggarly, three-suited, hundred-pound, filthy, worsted-stocking knave", "Before my body, I throw my warlike shield. Lay on, Macduff,And damn'd be him that first cries, "Hold, enough!" Role playing is there to make you really care about your defeats and triumphs. It's not some mere skill challenge to overcome.
There is a fundamental difference between physical and mental skills in an RPG - one that cannot be mitigated merely by providing mental skills. The body of the player may not be present within the game world so that they can use it in the game, but the mind of the game is unavoidably present in the game whether we like it or not. You focus on having the skill system provide skills to the character that the player doesn't have, but to a large extent it cannot do that. Even more so, the skill system is there to ensure that the character does not receive skills that the player has, but to a large extent it cannot prevent that. The problem with a stupid character in the hands of a smart player is regardless of what the numbers on the paper say, you can't remove the players judgment and problem solving skills completely. The problem with a high social skill character in the hands of a socially inept player is that you can't remove the poor social judgment from the play entirely.
If you were to try to do that, then you really would end up with a system where it didn't matter what the player said or did - the dice was dictating what the character did and the player was just acting out the will of the dice like some complex computer function receiving input.
Player: "I want to listen at the door."
DM: Make an intelligence check to see if you are smart enough to do that.
Player: "I slap the king."
DM: Make diplomacy check to see if he likes it.
Social and mental skills are inherently different than physical skills. Sorry to break it to you.