I don't think Shaxx has ever had any meaningful story involvement. As far as I can remember, his only non-Crucible dialogue after the Red War was the chat with Saint in the Devil's Ruin quest.
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Didn’t he have something to do at Twilight Gap in the lore?
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Yes, in the lore - but he has done nothing in the games and is very rarely involved in anything, not even providing throwaway dialogue in activities.