For that case, Crawford does bring up the example of when a PC is sneaking up on someone watching a play. You aren't obscured at all technically, but the DM might find that it is sufficient grounds to consider you to still be invisible. Very similarly what the other poster said about cases where you're passive perception isn't on because you're paying attention to navigating map. I took this a bit further, maybe, and said that every person wouldn't be instantly interested, or aware of every book in a room so as to pick out the false one.
What I think I'm seeing from you, is that you'd argue that in the case of sneaking up on the play attendent, it would still be possible to model this. Would you say something like that the quarry watching the play is just distracted, equating to some DC increase?
To me, noticing a book is "fake" is an example of the difference between investigate skill and perception skill. It's a difference between observation and examination and that is how I divide my "scenery" quite often.
Can an orc in a box be noticed without examining- yes cuz the box only blocks touch and sight but could allow sound and smell to give up the ghost. Depending on the nature of the box, it might provide advantage on the hide check or disadvantage to the perception roll.
For the sneak up on someone watching a play, I view that as an aid other "help action" by the play and so you get the benefits there to whichever side you want to apply the -5 (passive) or additional die (active.) However, if the individual had declared "watching for trouble, not watching the play" (like security duty) then that would not be the case but it would be rather clear they are not there to watch the play.
I also tend to do this with cases of "I'm busy" where I font say someone picking a lock or foraging fir food and game cannot use their passive perception to notice noises or smells or new breezes - instead I give them disadvantage on that passive perception.
I mean, especially in the outdoor activity cases, the idea that foraging, mapping the travel or navigation means you cannot use passive perceptions as a general rule seems a bit off since each of those involved direct study of and/or interaction with the environment that contains these things you might notice. I mean, if I am referencing our position relative to those two hills in the distance you would think I might be the first to notice the riders dust or storm clouds from that direction, right? If I am out trying to hunt down some game, I might be the first to notice the threat of another predator approaching our camp or signs of his recent activity.
The "choose a task" and that locks out other interactions seems at odds with using rules as resolving scenes with characters and more like a "pieces on gameboard" like say chess where if you are a knight this round you cannot go diagonal but if you are bishop you can etc.
There may be times when a strict division between action and awareness is called for, where passive perception is better not allowed because of other activities but for me the normal course of events and more commonplace cases would be for either normal perception/passive perception or for disadvantaged perception as opposed to locking out perception on a task basis.
It reminds me of GMs who insist you declare what you are searching for when you search an area or a desk or a pile.
THAT SAID it may help if your goal is to emphasize GAME and progression to use such a sort of "pieces and moves" style as it likely keeps things moving if one dispenses with some of that character stuff. Nothing wrong with that. I knew of a Hero RPG game where each session started with team assembled outside of crisis spot getting briefing by unnamed agents. It was done deliberately to get into the fight and action ASAP. I thanked them after one session and left telling them it wasnt the game I was looking for but they had a blast.
