For me the difference is in Dark Souls I was captivated by the world and felt there was lore hidden beneath the surface just waiting for me to discover through exploration. With Destiny, it felt like the lore was hidden because DLC.
I feel that. At least with Dark Souls, you could look at the descriptions of actual in-game items discovered through thorough exploration and pick up on pieces of the backstory from that, whereas with Destiny you need to read the cards on the app/website or whatever to get the big picture. Still, you're right, plenty of the lore in Destiny wasn't even revealed until the DLC came out.
There's some story there in the game, but you really have to pay attention or it would slide right past you, and once you complete the main story-line in the Vanilla version of the game (not the DLCs) you are basically left wondering what just happened. So yea, the Grimoire is a huge story buff to the game. If you really want to understand the things that are happening, you almost have to read it. If you are just wanting to shoot things and don't care, I wouldn't bother.
From my experience that's exactly how I'd explain the game. Everything is there to make something cool and grand but the execution falls flat and demands excess interaction rather than just telling the story through the actual game.
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