I consider "the door is closed, wait for it to open and fight lots of enemies" being "Horde Mode". I don't know the exact definition of Horde Mode so I simply went with that :p
And as long there isn't more enemy variety (and loot variety, i.e. more weapon stats which are randomized) and there aren't procedurally generated areas I don't think I'd enjoy a Horde Mode Arena for long.
I do agree that Destiny needs it, but a simple arena + random enemy spawns...won't do the trick.
But, I'd be happy if Bungie surprised us.
Edit: Something akin to Bloodborne's Chalice Dungeons would be incredible.