Terry Justice called the IT "a side show to a side show". In the context of the American Civil War, that's about right. Some of the more aggressive Union officiers wanted to conquer the IT but there were political reasons not to and, in the bigger picture, it was not a priority. If you're going to control the Mississippi, blockade the south, capture Atlanta, defeat the Army of Northern Virginia, etc., all you need to do in the west is keep the politicians and loyal citizens of Kansas and Missouri safe and happy. And that, in a VERY general sense, is what the North did.
Part of keeping the Kansans happy meant getting the Indian refugees out of Kansas and back into Indian Territory. They didn't really want the Indian Nations as allies because their intent was to disposes the Indians of IT of much of their land as punishment for siding with the Confederacy and relocate all the tribes in Kansas there -- which is of course what they did after the war. Phillips was a thorn in the side of the politicians because he actually wanted to fight the enemy.
Drew's regiment defected to the Union because they never considered themselves "Confederates" and they hated the Southern Cherokee (Watie's followers). The two units had to be kept separated to avoid open warfare between them. Opothle Yahola (leader of the so-called Loyal Creeks) warned the Keetoowah (or Pins) of Drew's 1st Cherokee Mtd Rifles how the Confederates would use them, he was right, and they waited for their opportunity to defect which came in July 1862.
Supplies intended for the Confederate Indians, including those Albert Pike paid for out of Indian funds or his own pocket, were intercepted in Arkansas by Hindman and Holmes. It wasn't fair -- it never is -- but Hindman had nothing when he got to Arkansas and had to create a replacment army from scratch. His theft of Indian arms, equipment, and supplies undoubtedly saved Arkansas for the time being.