We know, for example, that people's bodies die after they encounter nightmares in the dreamworld, but it's never been implied that the soul was ever harmed. Birgitte existed as a spirit construct until pulled out into the real world where she started dying, but she was a different entity from Asmodean.

There is just no evidence that being in Tel'Aran'Rhiod eliminates a soul, and the Dark One's message seems to indicate that Asmodeans' soul was gone, not absorbed back into the reincarnation pool.

I don't think Slayer is an intuitive or obvious choice for various reasons, but chief among them is that he was a forgettable villain in Tier II or Tier III of the hierarchy of lethality whereas the Forsaken, even the weakest, have always been identified as Tier I opponents. It is quite a large step to go from harassing the Two Rivers and wolves in the dream world to killing a Forsaken, destroying that Forsakens' soul and being able to pull bodies into T'A'R. As far as we know, Slayer's powers are limited to travelling into T'A'R by himself.