Adam Goodheart argues in 1861: A Social History Of The Civil War that we should remember how fresh and unproven the American experiment in self-government was. If the Southern states were allowed to leave the Union, what would stop New York or Massachusetts from leaving a few years hence over some perceived slight or economic imbalance? The idea of Union was not as remote and abstract an idea to fight for in the 1860s as it seems to us today, after a century of thinking of the US as a single nation tightly bound together economically.
We know how the experiment turned out—with a strong continental nation—but at the time there was no such certainty.