it is always going to be a tough balance between 'stuff to do' and 'immersion' with this title. If you add anything game-y it'll feel more like a game and less like an immersive experience. if you add anything that feels like unnecessary busywork, it can do the same.
I think the popularity of the coming storm was the fact that there is comfort in having progression and clear goals (game-y ideas). By getting an object, you feel like you are accomplishing SOMETHING (even if you don't yet know what/why).By using that item to get another item, that same feeling continues.
If you never want to have the player feel accomplished, in control, empowered, or able to understand anything in the game (so they are truly confused, aimless and disempowered) you would need to eliminate anything that would compromise those aspects.
Personally I like that getting certain objects and items has a bit of a trigger and so changes the landscape of the nightmare. get the candle and the storm rages on, allowing for the environment to alter in how dark it is, the rain visible, audio cues, jump scares, etc etc in addition to unlocking an area of the house you originally couldn't go. Get the axe and after you wake up and grab it again, you are sort of forced out of the main hallway, no longer feeling like you might me alone.
If the game had MORE of that, either additional objects to locate and carry, or possibly progress the level of nightmare's events by how many paintings you viewed/dolls you viewed, portholes into cells you viewed, that would also give a feeling of progression through natural gameplay. Players are going to do those actions anyway, so take advantage of it as a game mechanic that can tweak the setting in some way. Then if they don't do it, fine, they can still progress. if they DO do it and try to be thorough, the nightmare slowly changes and gets worse and the monotonous repetition(i.e. feeling of aimless walking) is changed.
One method to do this would be that as you progress through the first few nightmares in the mansion, if you always check the paintings, progressively the setting has more and more cobwebs. Looking at paintings of faces depicted so clear, yet the character (and also the player) have no idea who they are, feeling as if their mind and memories are full of cobwebs. you already have a lot of tiles and assets for spiderwebs and stuff so potentially layering them on is possible, and would age the house and make it feel more and more "lost" in the webs of time. If you added a shadow spider or hinted at it a few times for scares, that would be perfectly ok.

The loss of ones clear memories, short or long term, is also a mental condition many gamers will understand (especially aging ones like me lol).
Or in the Asylum, the more you look into cells, the more you are invading on someone's privacy. As you check the cells and see the horrors within, the more often you will hear the loose nuts running around and screaming (and since you already cited possibly trying different ideas with the inmates, this could tie into that with a graduating frequency or difficulty with them). Having a broken message written across a few cells that indicates a bit of the branching ending could help compel players to keep checking cells instead of skipping them (and so making the game easier).
having gameplay elements that don't FEEL like required elements, and are there in a natural sense (actions most players will do whether you want them to or not) is most likely what the series needs. if you want to avoid it being a game and getting muddled down with things that break immersion, it'll require clever but soft-gloves stuff like this to really work.