What I also use Passive Investigation for, is for example recalling things. Maybe you visited a house of a suspect, you later can, when new clues come up, recall details from there.
For examples, I agree to the other comment that Sherlock Holmes is no good example because he also has a high passive perception..
So, think of someone like Adrian Monk (like in the TV series) when you think about it.
He could maybe been hit on the head by someone sneaking unto him from behind without noticing, because I consider him having a low perception. But on the other hand he could maybe even recall the name of the books that were on the desk of the suspect when they visited him. Or if they talk to him to tell the player “you recall very clearly that the ledger he said he worked on wasnt on the table” if he tells the player something different.
Because I disagree on the part on actively telling the player stuff he isn’t looking for. If a player thinks that a room doesn’t have to be searched, don’t give them away stuff they only would get trough a investigation check.
But hey.. If they later think “damn, I should have searched the room!” they at least can count on their passive investigation, for remembering only very few basic details (DC 10) or quite a lot more (DC 15) and so on.
Also, maybe giving them some ready-to-go conclusions on a high amount, after having them actively rolled. Like, if they investigate the window and the glass shards are on the outside rather than the inside, and the player has high passive investigation, tell them what that means (“so you immediately reason it was crashed from the inside of the house!”) instead of only (“you see a crashed window with a lot of shards on the outside”). Because its fair to tell players who have a low passive investigation in real life, but their characters have in-game, to use it for such hints.