The relation with the Brutes and the Elites was never too good to start with. However, the Prophet of Truth truly believed that the Elites were inferior to the Brutes. And so, when he had a good enough reason (the failure of the Elite Honor Guards to protect the Prophet of Regret from death), Truth replaced the Elites with the Brutes. The Elites were originally to be 'reassigned', but instead, they rebelled, and the Brutes became some of the greatest enemies of the Elites. Theoretically, if Truth didn't have the Honor Guard replaced with Brutes, the two groups could have co-existed indefinetely.
EDIT: GORDY KELLICK
The High Prophet of Regret was specialist with the Sangheili forces, as he was a commander for many of their fleets. Rregret would not have tolerated Truth's arragont replacement of the Elites with the Brutes. Truth had the Phantoms that would have saved Regret withdrawn so that Regret could die, and Truth could replace the Elites without insurrection. Many of the Covenant were loyal only to Regret, there to say, most of the Elites. Mercy, as Truth's personal lap dog posed no threat and was allowed to live... for now. Truth believed that while the Elites were stronger and smart than the Brutes, they were prone to causing agruments and this really annoyed Truth. The Brutes, which, blinded by faith and jealousy of the Elites, would have jumped at Truth's proposal. Tartarus, leader of the Brutes at that time, unsuccessfully tried to kill Thel Vadamee, Arbiter and Respondant for the Elites. When the Arbiter found out that the Great Journey was founded on a lie, and based on a miscommunication in the "Terminals" tried to sway the Brutes in his favour, against the backstabbing Prophet liars. Now, with Regret dead, Truth allowed Mercy to be killed at the hands of the Flood ambush that awaited on their private Phantom dock. The Brutes, refused to listen to their natural enemies, the Elites and followed Truth out the back door. The Elites and most of the Hunters and Grunt left the Covenant along with few loyal Jackals, Drones and Prophets. This is part of the back story. They were not in the game due to the possibility of the confusion it would cause. Only the bad Covenant were in it. The Elites are operatable with, however.