I don't feel the force of the distinctions you're drawing here.
The captain's trunk and his daughter's hope chest are just placeholders I came up with to make a point. I could equally have said, as I've also been doing, Place X and Place Y. Or I could say the widget is in the captain's house and the players look for the widget on the captain's ship. Or I could say that the widget is in the possession of the captain (ship, house, trunk, who knows?) and the players look for the widget in the cathedral's baptismal font.
My point is that, in 5e D&D as canonically run, the players cannot establish the stakes of we look for the widget in the font. All they can do is find out what the GM says in response. If the GM answers A retributive angel materialises in anger well I guess that's the GM's prerogative. But I would find it poor GMing to (i) do that and then (ii) be surprised that combat started.
I agree that the DM who has the retributive angel materialize should be expecting a fight, and that if there isn't some foreshadowing of the possibility it's poor DMing.
To me, there's a functional difference between saying the book is in a specific piece of furniture, and saying it's in a specific room. Part of it is having played with DMs (or read adventures) where it was clear the PCs needed to search the specific piece of furniture to find the widget. I don't think much of insisting the PCs search the wardrobe specifically in order to find the widget (barring some specific reason to look in wardrobes).
I think that over the course of the Great Widget Hunt, the stakes of finding it/not finding it are likely to emerge, so that while the players don't have narrative authority of the sort that lets them put the widget in the baptismal font, they have arguably established the stakes of the broader outcome (or at least they know those stakes).
I would think of a "story element" as (say) the book or other widget, the font, the trunk, the hope chest, the ship, the captain, the daughter, the PCS, etc. The impression I get from your accounts of your play is that you as GM largely control the introduction of these into the shared fiction - with the PCs as an exception.
And I have been using "story elements" to mean things that happen in the story. The widget is a prop, or maybe a McGuffin; NPCs aren't really "story elements" most of the time. Very much most of the time, what happens in the story is dependent on what the PCs do. Yes, I introduce what you're referring to into the game, and I also from time to time instigate story arcs, though after the first one or two I try to have multiple arcs instigated and allow the players/characters to choose among them, but I also encourage the players to have a story arc or two pending in their backstories.
It is possible to have the introduction of story elements connected to action resolution. For instance, in Apocalypse World if the player succeeds on a check to have his/her PC read a charged situation then s/he can requre the GM to tell her which enemy in the situation is the biggest threat. This obliges the GM to introdcue a new element into the fiction - namely, one which explains why that enemy is so dangerous.
In this sort of RPG, the player's have more responsibility for what happens in the fiction. Eg if a player declares and succeeds in an action to learn who is the bigget threat, s/he can hardly complain if the GM starts telling her about the heavily armed bodyguards!
Funny. If a player in my D&D games asks about what's the biggest threat, there'll probably be a heavily-armed bodyguard (whether I'd intended that or not).