So far my conclusion is that the administration favored removing Rosecrans. Otherwise, why give Grant that option? And if you give him that option, why restrict the commander he could select to replace him with to Thomas? Simple answer, the resulting substitution of Thomas for Rosecrans is exactly what the administration wanted.
As to why not do the substitution himself? That's easy, Lincoln was trying to shield himself from political repercussions. Rosecrans was a known Salmon P. Chase protégé, Chase was a Lincoln adversary, and Rosecrans was a popular Ohioan war hero in the middle of a combustible political situation with Vallandiham creating troubles. Lincoln tread lightly, fearing the political backlash. Thus, he let Grant make that decision (knowing in all likelihood what it would be). In this way, the decision was left to a general who made it based on military necessity, and the political axe fell on someone else — who didn't care for politics at this point in time.
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Lincoln could have ordered Grant to Chattanooga to fix the situation with whoever was in command. Grant normally worked with what he had, and rarely asked the President for anything. In those cases he did, it was promotions for officers who had earned them and who he trusted — a reasonable request considering the level of discretion and authority Grant had and the kind of tasks he'd be delegating to these subordinates.
Exhibit 1. I present to you the case of Joe Hooker, who Grant distrusted, calling him a "dangerous man." For his own preference, he would have sent Hooker back. In fact one of Hooker's subordinates refused to serve under him, which created problems Grant had to deal with, without being able to remove any of them. Grant worked with a man he distrusted and used the resources he was given.
Exhibit 2. In the list of commanders appointed by Lincoln who had vendettas against Grant, I present to you: Gen. McClernand. This particular political general was the senior-most subordinate to Grant, which created problems for everyone else, specially Sherman, who Grant considered his right hand. Yet, McClernand proceeded to undermine and slander Grant every chance he got and had the President's ear for his vainglorious accounts of his abilities. It's evident he was seeking Grant's removal for his own appointment early on, but when Grant proved a formidable rival he couldn't dislodge, he switched tactics to ask the President for an independent command.
Grant worked with him even after Lincoln created a problematic situation promising McClernand an independent command in an area that was under Grant's control. Lincoln completely undermined Grant here, yet Grant politely managed the "embarrassment" of the situation as best he could. Conducted an entire campaign this way and only relieved McClernand when the general proceeded to "hang himself" violating regulations concerning the press.
Exhibit 3. In spite of his personal preferences, Grant was a subordinate soldier who worked with Lincoln's choices of commanders. In 1864 Grant was compelled to rely on the questionable abilities of Banks, Siegel and especially Butler for key roles in his campaign against Lee. None of these he was free to remove because the President held that power and he had political reasons. Only after the election did Grant remove Butler for justified military reasons. Siegel was removed after his failure in the Shenandoah, with the President's consent. Even then, Grant wasn't free to appoint who he wanted and Gen. Hunter was approved and decided upon by the President. Only after giving full opportunity to the President's choices, and all of them failing did Grant get his protégé promoted in Sheridan.
Lincoln also afterwards rehabilitated and reinstated McClernand and Rosecrans to commands, even though he knew Grant would prefer not to have either as a subordinate.
Anyways, I had to cite these examples because this theory that Grant only worked with his favorites is overdone and simply incorrect. He would have worked with Rosecrans if that was what he was ordered to do, but if given the choice, why would he choose Rosecrans?
Stanton gave Grant a personal briefing that described in detail the situation at Chattanooga, and the battle of Chickamauga as reported by Dana, together with Lincoln's appraisal of the stunned duck hit on the head. In his memoirs Grant recalled learning that Rosecrans had been run off the field with a large portion of his army, while Thomas put up a strong resistance and retreated in an orderly fashion. This was the reason he cited for the removal. It was a justifiable military reason given to him by Stanton. The fact Rosecrans had plotted against him, and he couldn't trust him was probably the icing in that cake, but as shown by the McClernand situation and other examples I cited, mutual dislike wouldn't have been enough to remove him.
Bringing the