What I think Bush lost faith in was Trent's ability to function as Majority Leader. His failure let Bush, and the rest of us, down. This shouldn't have been more than a weekend tempest but Lott's handling of the matter was amaturish. I have no doubt but that when this first happened Bush called Lott and told him to get out there and get this cleaned up. Lott probably told him he'd take care of it and then made a few phone calls. One to Sean and the other to Larry King. The fact that he thought that was sufficient says a LOT about his political radar.
Remember Lott had already let Bush down by telling the media that he wasn't going to bring up anything for a vote until January after President Bush went out and almost killed himself telling the people how important Homeland Security was. Lott just told the media, Aw shucks, we'll get around to it in a couple of months. The fact that President Bush had to come out publically and state "NO, you'll do it NOW!" was the first example of Lott's political tin ear. This fiasco was just the topper. Bush hasn't said he wants Lott out of leadership, he has merely stated he won't back any republican senator over any other republican senator on their decision for their own leader.