I'm glad you are even considering this as an option.
I personally like intrinsic more than extrinsic, but I do see where both have their merits. In terms of awards/trophies/achievements popping up that HEY you found a rock! "Rock Collector!" I do find it kills immersion and reminds you that you are sitting at a machine playing a game and not really "there". I rarely get 100% on games and in most cases, could care less how much of a completionist I am. The only game I felt bad missing a few things was "Never Alone" because the collectibles in the game unlocked documentary videos and the fact I missed 2 really bothered me as I loved seeing said videos.
One game that gives you a bit of BOTH and doesn't necessarily destroy immersion was Fallout New Vegas, where exploration allowed for a lot of environmental story telling, chances to discover people/quests/items/areas but also as you proceeded and got some of those achievements for say, 10 headshots, crafting 10 items, killing 10 bugs, etc etc, for every 5 you got a chunk of EXP to help you level the character, which I found a nice reward (especially if you are struggling to survive). Since the UI of the game was an extension of the PipBoy, the alerts and notifications never bothered me while I played. If I got a DING achievement on a console where it's the GAME SYSTEM telling me "good job" then I'd be annoyed. I guess thats the ultimate "immersion killer" for me- having the operating system of the hardware i am on (steam, Sony's software, xbox's UI) chime in really burns it.
So I guess extrinsic rewards that work within the game and it's framework universe and serves some function beyond "collecting it all" is ok with me. But I still am more intrinsic focused, I want to feel rewarded in a more subtle, organic way that is not so explicit and "here's a badge!" Environmental story telling, for me, is AWESOME and I play pretty slow, likely a lot like RightClick mentioned. So I love to explore, savor, and peer at every detail possible. IF a game has that in mind, and leaves breadcrumbs around for those people to reward them and expand on the story/lore/history then GREAT(especially if said items make sense in the world and don't feel thrown in/tacked on).
In terms of a list people can view to see if they found EVERYTHING... I can see the dilemma and agree it can be distracting. One possible solution is to have a bit of text or stats on the loading screen between nightmares. Like if you have a "chapter title" screen to let the player know they're moving to the next bit of story, also include stats on number of times died, time to advance, secrets found, enemies avoided, etc etc. In the main menu settings for gameplay you can add a checkbox to display or not during those loading screens so those that aren't interested in seeing it can leave it off. For those that are curious, or need to know, it is there (during a natural cut in action that is by its own existence, slightly immersion breaking) and if folk don't care it can be left off and just do a cumulative stats at the end of the game after all credits roll.
or if not all the stats, just a single stat. It would be kind of ominous to have a creepy font number fade in and out during the chapter/title screen loading showing the total number of secrets not yet discovered but with no other indication there (so it is cryptic and folk don't know why it's counting down, what it initially means). Eventually people will figure it out and word will spread, but only really to those that really want to know.
People won't know if it's hours until they die, mistakes they made so far, or some other gameplay element they're not sure about. hehehe. In this case I like the unawares feeling it can give and how it COULD be useful to completionists.
in regards to the significance/meaning of osme of these found objects/logs/notes/whatever and the game ending, and if gamers will GET the significance of the item first playthrough, or only in the second after seeing the end, I think it would work in layers.
For example:
First playthrough you are exploring an old shed and inside it you see a slightly out of place, well taken care of yoke for an ox on the wall. It looks like the leather is well cared for, there is no rust on the metal bindings, and it seems used but not deteriorating like much of the rusted dirty equipment in there. The play sees this and may come to a few conclusions about the type of farm this land used to be, that the farmer had cattle or some type of livestock to help handle the work, and that someone must have been there recently to clean or uncover the yoke.
Second Playthrough with the revelation of the family's secrets, you discover that the yoke was actually used to punish the farmers' family (kids, teens and wives), where they had to pull it and plow the soil for hours in the hot sun, and the farmer obsessively took care of these implements because they allowed him to do "God's work" and punish the wickedness out of his family. Along the way, through the game, any other "well kept" items you discover around that area also would gleam that added creepy realisation, and so second playthrough there is a whole different, possibly more sinister vibe to these discoveries.
The idea of "layers" can afford a lot of meaning and benefit to discovered items. A note talking about a man's sprouting love for a beautiful girl can later be read understanding he was a murdering rapist and was actually writing it while stalking someone's child. What was at first a lofty, almost romantic note that makes you sympathize with a character becomes completely twisted and horrible when re-read later.
I would love to see stuff like this implemented, but considering the potentially limiting budget/time/capabilities it wouldn't work if you, say, did lots of notes, journal entries, voice recordings and the like. Those are the typical staples of games nowadays and kind of washed out in my eyes. Using a FEW isn't bad, using it as a game long crutch almost exclusively though, would be painful and unimaginative.
Simple found images can sprout the imagination (or paranoia) of players and keep them engaged as they try to put together the clues. How abstract or subtle they are is up to you, how explicit the meaning is also up to you. is it obvious what it means, or will people only gleam the REAL truth after finishing the game?