Passive checks are for active PC's...but at the DM's discretion
This is a strange post for me, in that I don't think I have before now read anything by SevenSidedDie that I considered wrong. A different interpretation than I would make, perhaps, but valuable to consider and certainly not flat-out wrong. And yet here we are, with their accepted and highly-upvoted answer containing a mix of what is RAW absolutely true - players should never call for checks - and absolutely false - 'passive' checks have nothing to do with PC's being passive.
The claim (emphases in the original):
The operative word is passive. If a player is actively searching/examining/studying/watching, it's active, not passive. If a PC is actively using a skill, they roll for it instead of the DM using a secret passive check.
The only exception to that rule is the one you quoted: the DM can use a passive check to find the average result of doing the same thing over and over again, regardless of whether it's active or not.
While "repeated tasks" are indeed a time when a DM would use a passive check, they are most certainly not 'the only exception to the rule that active PC's make rolls'. To see this, we merely have to read the section of the PHB that defines passive checks:
A passive check is a special kind of ability check that doesn't involve any die rolls. Such a check...can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster.
An active check is appropriate when the PC's can immediately see whether or not they are successful - if they leap over a particularly wide chasm, they will know they are successful on a high roll and will know they fail and fall on a low roll. But when what they are trying to do is based around knowledge, so that when they fail they don't know that they failed, a passive check is often more appropriate. Passive Perception is used for searching since often the PC's will have no idea whether they failed to find a monster because one was not there, or because the one that was there was so well hidden. If you allow players to roll, you are giving them meta-information that their characters don't have.
Active Perception is not when a character 'actively looks' for hidden monsters, it is when the player actively makes a Perception roll (because the DM told them to).
Passive Perception is not an intuitive sense the characters have even when they don't search, it is when a character actively looks but the DM determines the result by using their passive Perception score without the player making a roll.
Any ability check, active or passive, is a result of the character's activity and conscious intent; as described in the PHB's section on Ability Checks (emphases mine):
An ability check tests a character's or monster's innate talent and training in an effort to overcome a challenge. The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure.
Passive ability checks are included as a subsection of the Ability Check rules and must follow the general rules for ability checks; they happen when the character is making an effort for them to happen. Further, it can be assumed that most of the time your character is on the alert for danger. As the Hiding sidebar (PHB 177) says:
In combat, most creatures stay alert for signs of danger all around, so if you come out of hiding and approach a creature, it usually sees you.
The only time a character is not actively looking for danger is when they focus their attention on another activity. And when they do so, they don't then use passive Perception. Rather, they lose the the ability to make passive Perception checks, as described in the PHB section on Noticing Threats:
Use the passive Wisdom (Perception) scores of the characters to determine whether anyone in the group notices a hidden threat...Characters who turn their attention to other tasks as the group travels are not focused on watching for danger. These characters don’t contribute their passive Wisdom (Perception) scores to the group’s chance of noticing hidden threats. However, a character not watching for danger can do one of the following activities instead...
When the characters are actively looking for danger, but the DM does not want them to roll, they get a passive Perception check. When they stop actively looking for danger, because they are engrossed in something else, they get no check at all.
'Active' and 'passive' refer to what the player and DM are doing in the meta-game (rolling or not), but not what the characters are doing within the narrative. The DM should not be determining whether the players 'get' an active roll based on what they say their characters are doing, but rather on how the DM wants the meta-game to proceed. What the players say the characters are doing is the basis for permitting a check in the first place, regardless of whether the DM decides it will be active or passive.
From the OP:
I could see if a passive insight check beats a deception roll, the person realizes they're being told a lie, without first having to say they're trying to use insight on the person. While you lose the part of the game of player's having to carefully consider each part of the conversation to see which parts they want to call insight on, some player's could just state "I insight them every time they tell me something.
If the PCs are suspicious of an NPC, you can assume that they are actively using their Insight on every unverifiable statement. The DM could say "Roll an Insight" after any particular statement, but then if the player rolls particularly high they might conclude 'He can't be lying because I rolled a great Insight', while if they roll low they can think 'I don't believe that, but I have to pretend my character does'. Instead of permitting this meta-knowledge, the DM could just allow the NPC's statements to be met by the PC's passive Insight - and while obvious falsehoods would be detected, particularly skilled lies would go undetected, without the players having meta-knowledge about it.
From the OP:
Is it they walk into a room with an obscure holy symbol on the wall, their passive religion checks beats the DC to recognize it and "as you walk in, you see a symbol on the wall and immediately recognize it as the holy symbol of 'so-and-so'," or do you wait until they state that they want to examine the symbol?
The DM describes the symbol as the characters walk in. If the players ask whether their characters recognize it, if they take an active interest, a check is appropriate. The DM might permit an active roll - but here again, this allows the players to know that they likely failed, when their characters would not. Instead, the DM applies a passive Religion check and simply narrates the results - and the players are left to wonder whether they are getting accurate information about a well-known faith or false information about something so obscure their characters are simply misled.