[D&D 5E] Passive Investigate, and Investigate as "Deduction and Memory"

I've been having a tough time conceptualizing what Investigate is for in 5E, and thinking of it as "Deduction and Memory" has finally made me feel better about it. Here are some outside-the-book applications I've thought up:

The first and cleanest use of Investigate is as the proficiency that applies when the player goes "dammit, what was that merchant's name again?" and the DM says, "roll int to see if you can remember."

What is passive Investigate for?

Your DM can cater descriptions to fit your party's passive investigate scores, and your party can use investigate actively to piece together particularly puzzling pieces of the story. As DM I sometimes wonder how generously to describe a scene, and passive investigate scores solve this.

The difference between low and high passive investigates is something like this:

[low]There's a man lying dead on the ground in the northeast corner; the door to the west is broken down.

[high]The man was trying to hold the door shut, but something burst it in, killed him, and tossed him across the room. Based on the wounds, it looks like it was probably one of the trolls you've been hearing about. Also, you realize the man matches the description on a wanted poster you saw back in Waterdeep.

Without investigate, if the players don't think to ask about the man's identity or circumstances, they miss out on information. But characters who are trained investigators (or just intellectually sharp) can pick up on these details automatically. Things like NPC motives or story connections that only make sense if you've been paying close attention can be given as freebies if somebody at the table is role playing Sherlock.

That's what passive investigate is for, and that's the other half of the Observant feat, which I know has been causing a lot of internet head scratching.

Investigate vs Perception

Finally, Investigate, by the rules, can be used to find hidden doors and treasures, which overlaps with perception. In my mind, investigate determines what a character can deduce about their surroundings, while perception determines whether they notice obscure details. Perception could tell you that there are two candlesticks on the mantle, and maybe even that the left one looks cleaner than the other; Investigate could tell you that the candlesticks aren't symmetrically placed and that the left one could, based on its position, possibly be attached to a latch lever inside the wall. You could find the secret door with either skill, passively or actively.

Based on these realizations, I'm going to start using a lot more skills passively too, especially Arcana, Nature, and Religion.