I had to read this like five times before my head stopped being filled visions of a bizarre cow-man hybrids with cheeky grins.
For me, rolling happens only after everything is played out and an outcome is uncertain. If, during an exchange, it becomes obvious what the outcome should be, no rolling is necessary. Everyone's mileage on "role vs roll" may vary, of course, but this way, I find it keeps the absurd results to a minimum while still leaving the players with reasonable agency through roleplay.
I think this is an approach that's true for almost all DMs, but to a some severely greater or lesser degrees, and that's where some conflicts can develop.
I mean, I had a DM who was so extreme that he wanted to eliminate all social and mental skills (and all social stats, mental stats merely measuring your ability to do magic), except as background elements, and this, I felt, was really a problem because it meant that a player who wasn't super-adept socially, who wasn't great at arguing on his feat, but who could come up with plausible approaches his character could take, was basically locked out of succeeding at anything social, because he couldn't actually act it out well (even though he could describe what his PC was trying to achieve). We explained to him that he could do that, but he would be DMing for a group of zero, were that the case, which sorted that out, but he frequently attempted to suggest that unless you could RP exactly and precisely what he wanted (which was almost pixel-hunting in some cases), you couldn't even roll, let alone succeed.
I've also directly experienced other DMs who used "we only roll when it's uncertain" as an excuse to just blockblockblockblockblock player ideas and RP that they didn't like - no matter how well-RP'd or reasonable their approach was. I've seen this a few times, too, which makes me leery of strongly advocating for "roll only on last resort"-type deals.
I think it's pretty clear that, in reality, most DMs are firmly in a middle ground where they won't allow a lucky roll to do something that is completely unrealistic, but won't block rolling unless detailed RP takes place, or only allow it when they, personally, are unclear on what would happen (i.e. have no specific opinion). Plus, if the roll does take place, I think it's really bad behaviour to ignore it - it might not get the PC what the player wants, but if the PC is trying to convince the king to abdicate in his favour, a natural 20 + big skill bonuses will mean the king laughs it off and maybe even considers the PC to be witty and daring (he might even grant him one of those weird medieval court titles), and generally improves in his opinion of him, whereas a poor roll would result in said PC ending up in the stocks, or worse, the dungeons. I don't think it necessarily benefits the game to try and RP out the entire conversation, and it can actually be unfair, in that RPing everything favours people who are fast-talkers, imaginative, and think on their feet (i.e. people like me), whilst penalizing plotters and planners, even if they come up with very theoretically clever approaches to the situation, and even if - importantly - their PC is someone who has the gift of the gab, or is supposed to (say, High CHA, Diplomacy, Bluff, etc - that stuff cost the PC - it is thus important not to ignore it).
My strong experience, too, is that it is really, really easy to deal with the results of ridiculous rolls, and more fun to to that, than to just say "Nope". Sometimes you have to, of course.
As an example of when one does say "Nope, sorry

" though, last night the PCs in my game were exploring the sewers of a city they didn't know well at all (none of them had been in that part of the continent), and they asked if they could make a check (I forgot what, something that would normally be appropriate) to guess exactly what was above them without actually going to the surface (they were under a part of the city they'd never been in or heard much about), and I had to say no, they couldn't get detailed information of that kind like that - a good roll could certainly tell them the general area, and maybe they could guess the trades, wealth level and so on from what was down there with them, but they couldn't just know the actual buildings, addresses and so on.