The north voted for the original thirteenth amendment which would have enshrined slavery forever in an effort to keep the south from leaving the union. Slavery was NOT why the south left. They also knew that war with the north would be futile so they did not want war either. The north did not want to see it's cash cow leave the union and decided that forcing the issue into a war was the answer to it's problem. This is the truth of what happened no matter how you try to spin your account. The north won and EVERYBODY lost.
EVERYBODY lost.
Maybe every one, except the 4 million slaves in the country that were no longer slaves. Bet they thought that they had something to celebrate.
Your basic point though is correct and rests on the better evidence of Lincoln's own statements: his goal was to preserve the Union, whether that meant protecting slavery or abolishing it. The South though insisted on slavery being preserved at the cost of the Union because they did not want to remain in any form of political association with states that carried the virus of any potential emancipation.
Most of the South thought that the Civil War would be brief due to the South's superior martial valor and skill. For the Southern public, this seemed not an entirely unreasonable line of thinking in that most of the federal officer corps went with the South. The Northern public was not much more realistic.
The esteemed but elderly Gen. Winfield Scott knew better. He proposed a multiyear strategy that raised and trained armies for the North, with the South to be blockaded and slowly weakened. Eventually, the North's armies would invade and divide the South and subdue its populace. This was of course how the war went, but few except top military professionals saw that coming at the time.