I'm not an extremist either. I can't "roleplay" for crap, that is, I am unable to "immerse" into a character (I can barely "immerse" into my real self), but I can and do have fun with "traditional" games. I've just "been there, done that" to some degree. Also, I don't begrudge character immersion as a goal of play, though it's a rather insular goal that the other players can't really share with you. It's all a matter of taste. Perhaps if I had any ability to immerse in characters, I'd have never grown bored with previous games I've played. Who knows? And of course, as long as everyone is having fun and not creating bad habits, what's the problem?
That said, a couple of points.
Do the combat rules "get in the way" of combat? Why would roleplaying rules "get in the way" of roleplaying? They might break immersion, but so do combat rules. Combat might be complex and have high stakes, requiring some form of system to structure it, control it and abstract all of the little variables that the players can't see or whouldn't know about. Of course, a debate, argument, seduction or whatever might be complex and have high stakes, requiring some form of system to structure it, control it and abstract all of the little variables that the players can't see of wouldn't know about.
Also, while players will accept that they can't reliably attack or defend in combat, that they might not be able to move how they want on the combat map, that they might have to choose from a menu of options when their turn comes up, they refuse to accept the same restrictions when it comes to "roleplaying". This is understandable since the character's decisions are the only thing that the player really has any control over, but it's certainly not "realistic". People are minefields of habits, expectations, misconceptions, indoctrinations and hormonal surges. That's not even taking into account the social skills that a PC might have that a player might not (or the other way around).
Also, while I suppose immersion is a worthy goal, it's kind of a sham. There's no real adrenaline rush from combat, no surge of irrational hatred for the man who killed your parents, no confused understanding while drunk, etc. The player can probably approximate it, figuring the odds and rolling the dice in his head, but that is, despite being hidden, stepping outside of the character. And even then it's likely a calculated risk, tempered by the player's squeamishness. And again, that's ok. We're there to have fun, so there's no point doing things that aren't fun. However, I doubt most immersive roleplayers would admit to being "all about the immersive experience, unless the character actually feels a genuine emotion or feels bad about something".
And finally, it's shorthand. Sure, you'll decide how your character feels about stuff, but you don't have three weeks of real-time to mull it over. The baron might be able to convince you to betray the queen (for the good of the kingdom, of course) after an hour-long debate over cognac, but the other players don't have time for that and likely aren't that interested anyway. If you want to get anything else done that night, you don't have time for that. So you go to the dice.
"But my guy would NEVER go for that!" you might say. How would you know, unless you roleplayed it out for that hour and really, unless you actually were that character? And surely the baron's controller thinks that his argument was undeniably persuasive and that the other player is just being obstinate. Now, despite the lack of rules (maybe even because of it) there's a disagreement about "what should happen next". And really, that's what a good chunk of rules are for, to determine "what should happen next" when the participants obviously disagree or just don't know.
That's another reason to follow the rules of a well-designed game. To do otherwise is to inflict your will upon the other players, regardless of their input and agreement.
Yes, "refering to the referee" is a perfectly valid resolution method. Refering to the rules and then ignoring them is not.
Basically, to get back on topic, "The DM can choose to ignore or change rules on the fly" is broken. However, "The DM can choose when and how to apply the rules" is not (necessarily). It's a subtle, but I think important, distinction.
"If you fudge dice rolls, your system is failing you!" -Sons of Kryos
"I genuinely think that some people measure their free will by reading how unhappy their GM is." - TonyLB