Per
Planet Destiny (scroll to the bottom), yes the strike-specific rewards can be obtained via the director version. No one has any clue re: drop rates for each version: director vs. level 36 vs. Heroic. But I gotta figure it's gotta balance out in favor the director since you can run that over and over and don't have to do other strikes.
That site is down, Epyon. The problem I've noticed is that a lot of people assume, as I initially did, that because
some strike exclusive gear drops in the director all of them do. The Dust Palace is a fairly unique case, though. It's an old strike that can still match you with vanilla players and the loot reward method is not the same as the vanguard and heroic versions of the mission.
It's fairly easy to find people who've gotten Taniks and other gear from the director version of their respective strikes (I've gotten three!). If it were possible to get the Mantles in the lvl20 strike I think we would encounter more people who would likewise chime in and confirm. Of all the players I've messaged in the Tower over the past two weeks (yes, I message every Mantle wearer...), all of them got it in the heroic/vanguard strikes. This is consistent with the drops for GAFfers as well, yeah?
Until there's actual proof it can drop in the vanilla strike I wouldn't suggest flayer farmers waste their time.