What isn't yet clear is whether or not he'd be able to exert the same authority over the other races of the Horde.
Well, Tauren and Blood Elves I think would be a no brainer - He was a close friend of Cairne, appears to be a good friend to Baine, and he brought his men to defend Quel'thalas and defeat Zul'Aman.
Goblins are trickier. Garrosh seems to push them around rather easily. I'd imagine any Warchief would have the same authority, even if they acted differently with it.
Orcs and Forsaken would be the trickiest. Orcs, I suppose it would depend how many are still of a Garrosh mindset after the Siege. Any Orcs that think like Thrall, Saurfang, or even the old Shaman in front of Orgrimmar shouldn't have any problem with it.
As for the Forsaken, I'm always looking at Bwonsamdi on this one. If he indeed has power over all types of Undead, and not just Trollish Undead (Which is pure speculation), the Siame-Quashi should be able to hold them down. It would be like Lillian Voss and Darkmaster Gandling all over again.