Personally I think Y'saarj has been dead all expansion, without exerting influence over Garrosh. I think when Blizzard say Y'saarj is dead, they mean his will is gone, dead Old God is probably incredibly powerful/hard to remove stuff and whilst it can cause mutation etc isn't acting in such a way that is sentient. This is why Garrosh isn't corrupt and how he has successfully weaponised the Heart. The Sha whilst being born from Y'saarj are not actually Y'saarj reborn but completely independent entities with their own agendas etc.
I don't see why the Klaxxi would try to stop Garrosh, infact before someone posted the Dungeon Journal entry I thought they had just been recruited into Garrosh's True Horde considering that they have a shared interest in the relentless pursuit of strength and culling the meek (both your own and your enemies). I think the explanation about the Titans burying the Heart is probably spot on, I doubt any of the mortal races knew it was there......it's quite probable that not even the surviving Mogu knew it was there.