I believe you can find what you're looking for on youtube. Bungie posts a cutscene archive for each season where it's just a compilation of each cutscene. For the smaller things like character dialog during a seasonal mission or hologram dialog you can check out Destiny Lore Vault. Just browsing on YT, I even found a 1hr+ video from esoteric where it's a complete story for season of Defiance including all cutscenes, dialog, quests separated by each week. He might have more of those videos. Hope this helps.