Maybe the hidden thing is intended as another non-guaranteed path. If they don't find the secret door, they might figure out that the books are all the same age; likewise the other way around. Multiple paths to get to a goal, none guaranteed. If those things make the world and/or the NPCs seem more real or more realistic, that's at least a bonus.
A secret door you're guaranteed to find is not really secret, is it?
Also, I gather that some folks in this thread have never lost their keys. If you are one of those people, let me explain it: You were in the kitchen. You had your keys. You know both of these things to be true. But now you don't have your keys. WTF? How is that possible? You search the kitchen. No keys. You go to the living room. No keys. You go to the bedroom. Nope. You check the damn bathroom even though there's no goddamn reason they would be there. Nada. Frustrated, you go back into the kitchen. There they are. Right next to your wallet. What?
So, the idea that there is "no roll necessary" even when a player says "I search the room" or "I search the dresser" or even "I check every drawer" doesn't make sense. Maybe you just don't find it because you failed to see what was right in front of your face.
Now, someone is going to say, "That's not fun! Now the PCs can't move forward" or something similar that emphasizes the gameplay aspect of the key. Well, here's the thing: if the PCs can't fail to find the key because it is necessary, there is no minimum level of the much touted "reasonable specificity" because they MUST find it. Otherwise we are back to -- you guessed it -- pixelbitching. Which, if you don't know, is technically defined as The Worst Way To Play D&D Ever.
Now, all of this ignore context, of course. Where are we? What is the party doing? Is it a dungeon? IS it an active residence? Who hid the key? Why? What does it open? Who knows the key is there? Do the PCs know they are looking for a key? Can they at least guess? Context is everything and none of the other arguments happening in this thread can be answered until the context is taken into account.
For what it is worth, I think there are two broad categories involving the hidden key: it is a macguffin, or it's not. If it is, the PCs are going to find it, so it isn't really hidden. "Searching" for it is mostly about contextualizing the transition from one adventure stage (looking for the key) to the next (using the key). If it isn't, then it is entirely possible the PCs never find it and that's fine, even if they miss out on whatever treasure, encounter, story and/or "flavor text" it opened.