This is just foreign to me. Players asking to make checks are simply asking to interact with the fiction in a way that feels meaningful to them.
But they don’t need to make a check to do that. They could simply describe what their character is doing, which unlike making a check, does not by its very definition have a chance of failing.
If they ask to check for traps at the door, that declaration doesn't suddenly change the state of the door trap. It was either trapped or it wasn't.
Saying they check the door for traps is not the same thing as asking to make a perception check. It’s a bit vague as action declarations go; it tells me what the player wants to accomplish (find out if the door is trapped or not), but it doesn’t tell me what the character is doing to try and find out, so I would ask for a bit more specificity here. But this isn’t what I would call an example of a player asking for a chance to fail. It’s a player declaring a goal without a clear approach.
Sometimes when the player asks to check a door for traps, the door is not trapped and never has been. The roll is not necessary but it would break the fiction of the world and tension of the moment if I say "Don't bother, it's not trapped".
I wouldn’t say “don’t bother, it’s not trapped.” I would say what the player finds based on the approach they described. Which in this case would be nothing. I don’t need to call for a roll to know they find nothing if there’s nothing to be found, I can just narrate the results of their actions
The roll reflects an action taken by the PC, even if it was not necessary.
This is not how I treat rolls. In my games, the player states the action taken by the PC, so there’s no need for a check to reflect the action. A check merely resolves uncertainty in the outcome of the action the players declared.
Sometimes the door was trapped and opening it without checking, so it will soon be followed by a "make a ___ save".
Will it? I don’t know without knowing what the character did to check for traps. If they’re prodding at it with a 10-foot poll, they might not need to make a save to avoid the trap. Or maybe they will, depends on the trap. But the point is, I need to know not just what the player wants to accomplish, but also what the character is doing to try and accomplish it in order to know what the results will be.
But making up a penalty because they asked for a check? That's just weird to me and seems like the DM using a cudgel to enforce a "correct" way of playing.
No one is suggesting making up a penalty because the player asked for a check. I literally say as much in the very post you’re quoting. The point of the adage “a player asking to make a check is asking for a chance to fail” is
not meant to suggest you should invent a chance to fail that wouldn’t have otherwise existed if a player asks for a check. It’s just to illustrate the absurdity of asking to make a check (which has a chance to fail) instead of describing an action (which might or might not have a chance to fail).
If someone is wasting time checking for traps every 5 feet, it's time to have a discussion about pacing and not making repetitive checks. We discuss pacing and how we can just use passive checks but it will slow movement to a crawl; we discuss it openly as an approach to the game instead of punishing people for playing "wrong".
I don’t punish people for being “wrong” either. I also don’t need to have the pacing discussion because I use level design and telegraphing to give players clues of when and where they might want to check for traps.