Ah, the Corwin Amendment. The south rejected it because it wouldn't guarantee slavery in the territories, which was the platform Lincoln was elected on. Lincoln, in turn, rejected the Crittenden Compromise, which was basically the same except that it guaranteed slavery in the territories.
Ah, the Crittenden Amendment. Most of Lincoln’s base was Free Soilers, who didn’t want blacks, free or otherwise, anywhere in their states or territories. He had to oppose that. But it wouldn’t have made any difference if it had passed because even if it had passed the South’s main issues involved the American System (Lincoln’s and Henry Clay’s system of crony capitalism).
There is no mono-causal explanation of the Uncivil War. It is only sanitized schoolbook history that was common into the 70s that tried to force the conflict into the procrustean bed of “slavery vs. abolition”. Liberals historically have loved that narrative, too, although at present they are revising their views in ways intended to inflict ever more unfounded race guilt.