Nazgrim is just as much a tool as any of us player characters. He's a soldier. He does what he's ordered. That doesn't mean he necessarily likes carrying out every order. He said as much in Grizzly Hills when we met him the first time. "Oh, they sent you here? They must not like you very much." That sort of thing. He fights for the Horde's banner. Oops, so do we, and all the rest of the Horde including every one of the other racial leaders, including Thrall. He issued strict orders for his men not to harass the mixed-race crew of the ship that brought them to Vashj'ir. He fought tooth and nail to defend said men when they were swarmed by naga, to free them when they were enslaved. And the only time I've heard him mention Garrosh's name was to lament that Hellscream would take his head for getting the airship destroyed.
So really. Lay off the guy. He's cool.