Garrosh couldn't plan a bar mitzvah.
We're not dealing with a master of subtlety, here. Malkorok, perhaps, but not Garrosh.
That's the act he puts on so his enemies will underestimate him. There is a reason why he was able to advance himself to the position of Warchief so quickly, by manipulate that simpleton Thrall. Garrosh may come across as abrasive and racist, but he only sees in the horde races weakness that needs to be purged. Not with genocide but with words, get them riled up and angry, back to a stronger horde united under the strength of their own power.
Take for example the blood elves. They were atrophying and weak, rather pathetic and even ready to leave the horde. Garrosh knew of their eventual betrayal, but saw in them a power that could make the horde strong. Garrosh ordered his puppet Aethas to make the sunreavers to once again betray Dalaran. With Jaina at the helm and the previous betrayal of a sunreaver agent in the bombing of Theramore, he knew she would not forgive a second betrayal.
So now the blood elves are much stronger and angrier, even Lor'themar is bringing back the Spellbreakers and working on making the blood elves much more powerful, powerful enough to not even need the horde. This would have been impossible without Garrosh and they would have continued to languish forever as the pathetic weaklings they have become under the protection of the horde.