In the end, I think that Whitman seeks to focus on the horror of war and its unifying experience of death because such universality is a part of Whitman's poetic experience.  Whitman sought to create connection with his work.  Whitman's writing is the essence of connection with the world, with others, with the reader, with himself.  This thematic element of connection is challenged with war.  War, by definition, severs connection between individuals.  The reality of death is where Whitman focuses his poem.  For Whitman, the connective thread of war is death.  Regardless of the rationale for war, it is an experience where only death is evident.  For Whitman, this becomes "the heavy blanket" that covers all, connects all, and binds all:

Three forms I see on stretchers lying, brought out there untended lying,/ Over each the blanket spread, ample brownish woolen blanket,/ Gray and heavy blanket, folding, covering all.

It is for this reason that the three bodies upon which Whitman focuses are not necessarily "Union" or "Confederate."  They are of the father, the son, and the holy spirit. For Whitman, the lifeless bodies he sees are reflections of the life taken away.  This is a condition that unifies all of human experience.  The philosophical issues raised by the Civil War are secondary to the vase experience of death that is a guarantee with all war.  For Whitman, the rationale behind why the war was fought is a far distant issue in comparison to the reality of death that was forged as a result of it.  In this, Whitman concentrates on the experience of death that is such a part of the Civil War experience.