I finished the story with my first character, for the first time on Xbox One yesterday, and did not receive the Stranger's Rifle or play the "ending" cinematic.
I will say that I previously played through the game on PS4, with a PSN account that is also linked to my Bungie.net account, so I would guess that this is the issue, but I'm not certain.
Has anyone else played the game one two different consoles, and did the ending cinematic/Stranger's Rifle not appear for you?
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How many different consoles you play it on shouldn't matter at all. Try logging out then restart the game and just replay the last mission