I'll echo some advice I've heard before here on ENWorld: Why would a player ask to make a specific ability check in 5e? Given that the 5e DMG points out that the DM should "Only call for a roll if there is a meaningful consequence for failure", a player requesting to make a specific ability check is asking for a chance to fail and thus harm the party in some way. Wouldn't the better play be for the player to describe what their character is doing and what they hope to accomplish - perhaps invoking a trait or a resource or a past experience that makes them particular good at the task they describe - in order to suggest to the DM that they should auto-succeed at the given task? Or at the very least have the DM grant them advantage or lower the DC because of their described approach? Just seems like a smart play. That's not to say you can't have fun in a game where the players call for rolls - its literally just not how the 5e gameplay was designed to run. So yeah, Angry is not wrong (except - and this is a big except for Angry - in his schticky, blatant under- and overtones of "badwrongfun", which only serves to turns people off to what is often very sound advice):
Context is kinda key to that quote, especially the first sentence:
Finally, the Core Mechanic very explicitly spells out that the rules for action resolution are TOOLS used by the GM to determine the outcome of ACTIONS chosen by the players. It may not seem like a big deal – because we all know that’s how it’s supposed to be anyway – but that wording is very useful to new GMs and new players. And some experienced players and GMs need to hear that too. Under these rules, a player who asks the GM “can I make an Insight check” is not playing the game properly. They are playing against the rules.
Again, everyone should feel free to use the rules however you like as DM. But when that causes some issue or awkwardness at the table (like the cascade of player-invoked rolls to accomplish the same task mentioned previously, for example), just refer back to p6 of the PHB for the intended general flow of 5e play and to p237 of the DMG for the 5e designers' intent for how ability scores are to be used in play.
The player is declaring an action, just not an action that has any physical effect in the game world. They are actively trying to determine if the NPC is lying. I guess I could have the players state "Can I tell if they're telling the truth?" or "I study them closely looking for signs of nervousness" but it's just unnecessarily clumsy and requiring specific phrasing that communicates exactly the same thing IMHO.
As far as "punishing" players for asking for a check ... I don't. In the case of insight and a few other skills I use the better of passive or the roll.
Having a PC in the party who is naive and gullible is quite common in my experience. If a PC's flaw is to be gullible and they're they only one questioning an NPC, the NPC can lie their ass off. As a DM I need some indication and reminder that they aren't going to be suspicious.
So maybe a different question: a PC has the flaw "I put too much trust in those who wield power within my temple's hierarchy. " They're questioning someone superior in the hierarchy that is lying but is proficient at deception.
To me this is a scenario where I would like to call for an insight check, but I don't know my player's characters to that level of detail. Any other PC would have a decent chance to detect the deception but it's not guaranteed.
For me it's simpler and more fair to the player's vision of their PC to let them ask for an insight check now and then.