I just thought I'd throw in my 2 coppers on why I allow people to ask for insight, use passive insights or ask for checks with an example. As far as how we get to the point of relying on a PC skill instead of player skill, I really don't care. To me that's just a stylistic choice. My point is that I think skills like insight can be used to give a lot of hints and clues, it can add depth and richness to the game that doesn't rely on DM or player skills.
Scenario:
The group is questioning a suspect in a who-dunnit. Think typical police procedural with the cops interviewing the suspect, Franky Nine Fingers.
PC: "Where were you last night after dusk."
Franky: "I was at home as usual."
PC: "Any witnesses?"
Franky: "Nah, I live alone. Nobody but me."
PC: "So you know nothing about Jimmy the Nose dying?"
Franky: "Jimmy's dead? No I didn't know that"
So short, but simple (the conversation would obviously continue). From the text, there's no possible way of knowing whether or not Franky is on the up-and-up. So let's add in some details you could pick up with insight. But let's say Franky has a decent deception skill and whether or not the PC should pick up on subtle details is not certain
PC: "Where were you last night after dusk."
Franky: "I was at home as usual." [he's being sarcastic and condescending]
PC: "Any witnesses?"
Franky: "Nah, I live alone. Nobody but me." [shift's a bit in his seat, this question makes him uncomfortable]
PC: "So you know nothing about Jimmy the Nose dying?"
Franky: "Jimmy's dead? No I didn't know that" [not really surprised]
It's obvious our Franky boy is hiding something, questioning/actions ensue. He's probably now suspect #1.
PC: "Where were you last night after dusk."
Franky: "I was at home as usual." [he paused slightly after "at home", is he hiding something?]
PC: "Any witnesses?"
Franky: "Nah, I live alone. Nobody but me." [Franky glanced nervously at the captain of the guard. Why?]
PC: "So you know nothing about Jimmy the Nose dying?"
Franky: "Jimmy's dead? No I didn't know that" [Seems to be honestly surprised]
What's going on? He may not be a primary suspect, but there's something fishy. Maybe the captain of the guard knows something? Maybe he visited Franky? Or maybe Franky is be being 100% honest but suspects that someone is watching his house.
Or any number of other variations. Both of these scenarios are incredibly common fictional tropes. The point is that subtle queues may or may not be picked up.
It shouldn't be up to the DM to act this out or the players to pick up on it because then it's not the PCs doing the questioning, it's the players. I am not my PC.