I agree with your premise - the Cornerstone Speech and the Confederate Constitution are unequivocal that slavery was the basis for Southern secession and the formation of the Confederacy. For those unfamiliar with this speech, in March 1861, a few weeks before the attack on Fort Sumter and the start of the U.S. Civil War, Confederate Vice President Alexander Stephens gave a speech in which he said:

"The new constitution has put at rest, forever, all the agitating questions relating to our peculiar institution African slavery as it exists amongst us the proper status of the negro in our form of civilization. This was the immediate cause of the late rupture and present revolution. Jefferson in his forecast, had anticipated this, as the 'rock upon which the old Union would split.' He was right." [1]

If I'm reading your question correctly, you're asking, "how can anyone ignore this?" Stephens says in no uncertain terms that slavery is the direct cause of the U.S. Civil War. And I agree with that read. The only way I know to answer your question, then, is to explain how the "slavery was not the cause" argument came to be.

Today's Confederate defenders are ideological descendants of what is known as the "Lost Cause". The Lost Cause is a form of revisionist history that recasts the conflict of the U.S. Civil War as one about the ideals of freedom and states' rights, and seeks to show that the rebels were part of a noble tradition of liberty. The Lost Cause is a form of mythology that "tells us that the war was a mawkish and essentially heroic melodrama, an honorable sectional duel, a time of martial glory on both sides, and triumphant nationalism." [2] Removing slavery as the central question and conflict of the Civil War - which it most certainly was - was necessary when creating this mythology, and allowing for white Americans, North and South, to reconcile over the meaning of the war. The Lost Cause ideology emerged in the decades after the U.S. Civil War, but became more prevalent following the end of Reconstruction.

The Lost Cause mythology allowed white Americans to set aside the question of race and come together over the shared trauma of the war. There were reunions where soldiers from both sides came together to share their memories of the conflict. Often African American soldiers were not included in these reunions, or their role in the war was minimized. One such reunion occurred to mark the 25th anniversary of the Battle of Gettysburg [3], and another to mark the 50th anniversary of the same. [4] These reunions - and many more like them - required a willful forgetting, a rewriting of history that allowed both Northern and Southern white soldiers to be men of honor who fought for a worthy cause. But it also meant setting aside the issue of race in America. The Lost Cause was, and is, a mythology that allowed the people of the United States to carry on without addressing the fundamental issues of racial equality in America.

When today's Confederate defenders argue that the war was not about slavery, they are drawing on a very long tradition of forgetting. They are choosing not to see the evidence in the historical record in the same way that many Americans - some of them veterans of the war itself - have been choosing to forget for over a century.

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[1] Cornerstone Speech, Alexander H. Stephens, March 1861: https://www.battlefields.org/learn/primary-sources/cornerstone-speech

[2] 'The Anatomy of Myth' by Alan T. Nolan, an essay in "The Myth of the Lost Cause and Civil War History", page 12 (portions available through Google Books)

[3] The Grand Reunion of 1888: https://npsgnmp.wordpress.com/2015/10/16/the-grand-reunion-of-1888/

[4] "All May Visit the Big Camp": Race and Lessons of the Civil War at the 1913 Gettysburg Reunion: https://cupola.gettysburg.edu/cgi/viewcontent.cgi?article=1018&context=gcjcwe

Additional Reading on the Lost Cause: "How I Learned About the "Cult of the Lost Cause" by Mitch Landrieu: https://www.smithsonianmag.com/history/how-i-learned-about-cult-lost-cause-180968426/