Garrosh had many attributes, with pride definitely among them. Garrosh kind of epitomized all the greater Sha in his way: Anger, Despair, Fear, Hatred, Violence, Doubt, and Pride.
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Pride wasn't banished by Shaohao because it was part of him, as well; as per your description below he wasn't looking for it in himself due to his own pride in his actions and standing. It's also a recurring motif for the Pandaren themselves, both generally and specifically - they are often prideful, sure of themselves and their way of life. But that being said, pride is an attribute of a person, and not really a thing that can stand on its own (outside the Sha itself, perhaps). Pride was an element of Garrosh, sure; but that's not all he was.