Then how do you explain this?
"In the momentous step which our State has taken of dissolving its connection with the government of which we so long formed a part, it is but just that we should declare the prominent reasons which have induced our course. Our position is thoroughly identified with the institution of slavery -- the greatest material interest of the world. Its labor supplies the product which constitutes by far the largest and most important portions of commerce of the earth. These products are peculiar to the climate verging on the tropical regions, and by an imperious law of nature, none but the black race can bear exposure to the tropical sun. These products have become necessities of the world, and a blow at slavery is a blow at commerce and civilization. That blow has been long aimed at the institution, and was at the point of reaching its consummation. There was no choice left us but submission to the mandates of abolition, or a dissolution of the Union, whose principles had been subverted to work out our ruin." (Emphasis added; From the opening paragraphs of the Mississippi Declaration of Secession.)
Most of the people in the Southern Confederacy did not own slaves.
Roughly 1/3 of Southern white families were slaveholders. They were the richest people in the South and called most of the shots for the Confederacy. They also went to great lengths to try to convince the poor, nonslaveholding white Southerners that if negroes were free they would threaten the poor whites' social status and have sex with their daughters and sisters.
Secondly, he did not "push through" the 13th Amendment. He was killed before he could do any "pushing".
The Thirteenth Amendment was passed easily by the Senate on April 8, 1864, but it took a great deal of arm twisting by Lincoln to get enough Democrats in the House to pass it by the required 2/3 majority on January 31, 1865. (Source; see also McPherson, Battle Cry of Freedom, p. 839.) John Wilkes Booth and his fellow Confederate conspirators saw to it that Lincoln did not live long enough to see it ratified by 3/4 of the states, but Lincoln's everlasting moral presence made that a mere formality which culminated on December 6, 1865.