It does exactly what it says: it gives you runes on completion of strikes and matches in Crucible and Gambit.
No, it does not. It doesn’t give me anything. No runes, no coins. No nothing
Strange. Is your current character able run the Menagerie? I.e you are high enough level and caught up on any possible prerequisite stuff (I don’t know if there is any, since the season startup quest and the chalice is account-wide). Maybe ask in the help forums?
I still have to have to explore the new raid, which I haven’t done
That isn’t required - I haven’t. But you do know that runes aren’t inventory items? (Luckily, since anyone doesn’t have space anyway…) So, you only see runes popup in your loot stream after those aforementioned activities, and then you need to inspect your chalice to see the runes you have collected (and unlocked to drop!).