If Lincoln had announced in April 1861 that he wanted volunteers to end slavery, he would have failed. Probably more states would have seceded than actually did (I would think Maryland and Kentucky would have seceded, and maybe Missouri). Congress might have refused the necessary funding for the army. The people who wanted an immediate end to slavery, the abolitionists, were a small minority in the North.
The political elite in the South may have made statements to the effect that the war was about slavery, but I doubt the average Confederate soldier volunteered with the idea "I want to help defend slavery."