Brutes and Elites hated each other. Brutes are larger, and more powerful, yet are second class citizens in the Covenant.

As the war with humanity had drug on, some elites were beginning to question the prophets motives. Brutes were hungry and wanted power at any cost. And now you have the Heretic Elite causing a public scene and in possession of an Oracle. The Heretic Elite is in a much better standing to cause the Prophets political damage due to his status as a respected Elite and his possession of an Oracle. The Banished, controlled by a Brute, while a threat physical threat, does not have the political leverage and bully pulpit that the Heretic had.

Tartarus and the Brutes are more than happy to Usurp the Elites. It's a power/caste thing, and has nothing to do with the Covenant Religion. They are probably even aware of the Banished and secretly supportive of them. From Tartarus' point of view, he just hit the Jackpot on being the top Brute in the covenant.

Don't forget that the Prophets and Elites started out as enemies thousands of years ago, before stumbling upon forerunner tech and forming the covenant.