When most of that race backs the genocidal, racist maniac who uses the other races as cannon fodder and utterly shatters their trust in that race's ability to rule, that's when that race loses its place at the top of the power structure. That's the lynchpin of all this--the other Horde races no longer trust the orcs with unchecked power. That's why Vol'jin rebelled and why the others backed him up almost immediately. Hell, Lor'themar even went to the Isle of Thunder explicitly to look for something he can use to make sure Garrosh doesn't target Quel'thalas when Lor'themar rebels along with taking down Lei Shen.
The reason Vol'jin got the pick was because he ended up being the one pulling the revolution together and, yes, being smart enough to realize the rebels needed backup. His choice was between a three-way war between his rebels, the Alliance, and the Kor'kron, or getting the Alliance to back him and having the numbers to take Garrosh down. I know, crazy thought--someone uses their brain rather than their nuts when they plan a siege, and with an Old God's heart in his city, Garrosh presented a far more clear and present danger than Varian (who could at least be relied on to conduct himself with honor as an ally of convenience).