In peace, there’s nothing so becomes a man,
As modest stillness and humility:
But when the blast of war blows in our ears,
Then imitate the action of the tiger;
Stiffen the sinews, summon up the blood.
The title of Warchief is used to denote the leader of the Orcs in WOW. It's a term with a very clear implication: that the primary role of a leader of the Orcish people is to lead them into battle. Equally obviously, it's a role that Thrall has gone far beyond. He's a leader of the wider Horde, a spiritual leader through his shamanic links to the Earthen Ring and a subtle politician who sees the futility of the endless conflict between the Alliance and Horde.
From our external perspective it's easy to see Thrall as the perfect leader and very hard to see how anyone could possibly think Garrosh was fit to lead in the same way. I've made quite a few posts on the topic myself!
What's started me questioning this view is reading the introduction to Seamus Heaney's version of Beowulf, which places great emphasis on the cultural context the stories were set in. Beowulf's world is one where the lords and kings are, above all, war chiefs. They are the leaders and protectors of their people. The world is full of enemies and a ruler who cannot defeat or deter them is doomed. To quote Heaney:
"All conceive of themselves as hooped within the great wheel of necessity, in thrall to a code of loyalty and bravery, bound to seek glory in the eye of the warrior world. The little nations are grouped around their lord ... a lord dies, defencelessness ensues; the enemy strikes".
For all of Thrall's modernist consensus building, it seems to me that this is closer to the tribal Orcish mindset. Garrosh is of the "big dog" school of politics; it's not enough to be strong, you must also show show it to deter others. If Thrall is the ideal leader of a democracy in wartime, Garrosh is the ideal leader of a prison gang. Never blink, never compromise.
Garrosh is the kind of man who draws a line in the sand and dares you to cross it. If you do, you know he will kill you or die trying. If you don't, he'll cross it himself, because you must have been afraid if you didn't.
Is that the kind of leadership the Horde need? Of course not. His inability to compromise with his allies will split the Horde apart in the long term, and his inability to think strategically will lose him any war eventually. But in the short term, that hyper-aggression will win him some victories and his people will love him.