I like the character to be perceptive independently on the player. I think you're doing the player a disservice if he has invested in perception for his character, but don't allow him to find things that he as a player specifically may not mention. Just because the player may not think to notice the hidden trap door under the rug, the super perceptive character may still notice it.
All the player need do is describe what he or she wants to do e.g. "I search the room from top to bottom." The DM then narrates the outcome. In this case, I think a reasonable call for a DM would be to
at least give the character a chance to notice the hidden trap door under the rug, drawing upon the passive check or an active check to determine a result. I'd probably take it a step further and ask the player if they want to spend 10x the amount of time on searching for an automatic success. (But then my games generally have a countdown timer to Bad Things happening, so it actually matters.)
The point is, I think the player needs to at least make some effort to state his or her goal and approach clearly. Not only does this add vital context to build the scene, it makes it easier for the DM to adjudicate fairly.
You don't ask players to tell you how they attack a monster and then determine the outcome based on that, why gimp your PCs in the same way? Now if you like to DM like that, fine, but not all of us do, and I personally would not like to play under that style of DM.
Yes, I do ask players how they attack a monster (again, that context is vital for scene-building and fair adjudication) and then determine an outcome based on that. Generally, if a monster is defending itself, the outcome is uncertain and I call for an attack roll or ability check. If a monster is not defending itself or is unassailable, then the outcome is probably certain and I don't ask for a roll. I simply narrate a result.
Passive Perception doesn't work like "you always notice or do not notice set DCs". It's a way of cutting down numerous dice rolls. If the player active asks to search for something, then you make them roll (or in my case, I have macro's set up in roll20 to roll for them, I don't like them seeing their result). This way, they still have a chance of spotting things above their passive DC.
As I see it, if a player asks to search for something, I narrate the results if they are certain and ask them to roll when it's uncertain. If you use "progress combined with a setback," you won't have to hide the die rolls from the players. They'll always find the thing that's hidden - only at a cost or with a complication happening. On a failed check, the character might find that trap door hidden under the rug, but it draws unwanted attention or takes up more time than they planned.
Now in terms of stuff hidden away with high DC's. It's there to give players who invest in perception/investigation/etc a sense of accomplishment. These modules are written for a mass audience and a wide range of players/characters. Not everything is designed to be found by everyone. If you create your own content for your own group, your mileage will vary.
I think it's there to reward engaging with the exploration pillar. It's just that authors are listing DCs as a shorthand for difficulty -
if the result of the search is uncertain. If a character spends 10x the amount of time needed to search a room, they don't roll against that DC 40 safe tucked in the corner. They just succeed (DMG, page 237).