Reminds me of Wrath when Varian whined about how he was a forced to be a gladiator slave for the Horde, totally twisting the lore to make the Horde sound bad and to make him sound like a victim, when in reality the gladiator ring wasn't run by the Horde, he was only allowed to be taken prisoner because he had amnesia and was assumed to be a Theramore deserter, and most importantly, it's well established that he actually preferred the simple life as a gladiator to the complicated life of ruler.
Here, if anything, it was Cairne who betrayed Garrosh. Was Cairne wrong for having misgivings about Garrosh? No, not really -- though if you read the Shattering, it's clear that Cairne had already formed his opinion of Garrosh based on the actions of Grom, before he even met the orc himself.
Thrall appointed Cairne as Garrosh's advisor, and Cairne left because Garrosh wasn't heeding his advice, despite Garrosh practically begging Cairne to stay on as his advisor, saying that he respected his advice even if he disagreed with it. But Cairne left all the same -- and the issue of contention that forced him to leave, btw, was the architecture of the reconstructed Orgrimmar. It reminded Cairne of Hellfire Citadel. That was the tipping point for him to leave.
Then Cairne and Hamuul were deceived by the Twilight Cult. Completely fell for the bait they set up. Cairne didn't investigate, he didn't try to find the truth -- he marched into Orgrimmar, punched Garrosh in the jaw, and challenged him to a duel. Garrosh thought he could cow him by making it a duel to the death, in true traditional style, but Cairne surprised him by accepting. Garrosh and Cairne were then both betrayed by Magatha. Garrosh did not want to win the duel that way, as is made obvious by the book. He considered it a blow to his own reputation, and a robbery of a test of skill against another powerful warrior.
Let's not forget that Magatha was the true traitor, and that Cairne new she was a threat, but let her carry on under his nose. The Grimtotem not only assassinated Cairne, but killed many, many tauren in their attempted coup, and Baine himself would've been killed if not for a few Grimtotem that did not agree with Magatha's plot.
So, yeah, I have no problem with Baine holding a grudge against Garrosh, because Garrosh did strike the final blow, but come on. Let's at least phrase things accurately and not misstate facts just to make Garrosh look like a bigger bad guy.