“As it happened, the end of that institution was a fortunate outcome but not the reason for the war itself.”
Oh, I thought you had made the case the reason for the war was slavery. If you did not, many northerners who attended public schools do.
Let's look at your original question: "If the South was fighting for slavery during Lincoln's War, who was fighting against slavery?" You're right that the South was fighting for slavery. The South seceded to defend slavery. The South started the war. So the reason for the war can easily be connected to slavery. But you also asked if the North was fighting to end slavery. They were not, not directly.