Did Brandt wanted to kill Steiner?

I don't think so. Brandt reminds Stransky to contact Steiner's platoon after Brandt has started his investigation into Stransky's conduct during the first battle. It is also, after Steiner wanted 3 days to think about his testimony and Stransky knew this from Triebig. I have always interpreted this as Brandt being extra concerned for their safety because he knows that Stransky has a motive to do the unthinkable. Unfortunately, Stransky can simply say that the lines were cut and Brandt would not have evidence to contradict him.

Later on in the film, when Brandt finds out that the platoon was stranded, he looked genuinely concerned when he said to Kiesel, 'Do you think they can make it?'.

Given Brandt's overall portrayal, along with his reaction late in the movie, I think he was concerned for the squad.

I'd always thought that they had become so used to retreat after retreat with Steiner's platoon always left behind as rearguard that, Brandt felt the need to single out Steiner's platoon too just to emphasize the dire and hopeless nature of their situation.
Brandt is portrayed throughout as a father figure to Coburn's character and, although Steiner may be a wayward and troublesome child, Brandt in no way wants him to die a needless death. Especially one that would advance Stransky's career.