Why? They did uphold it: "The southern states would have never signed the thing if they didn't think they could get out of it!" --Shelby Foote. That was a legitmate interpretation by the southern states.
And I say this as a Yankee born, bred, and educated for most of my life in the North. As a kid I couldn't even stand driving through Tenn. & Ga. to get to Florida, and actually felt uncomfortable in a that foreign land until we crossed the Ohio River on the way back home. But even I can read and reason. It wasn't a "civil war" with 2 sides vying for power. It was a war for independence, which failed.
Most of the rebelling southern states didn't sign anything. The Constitution was ratified long before they were admitted. States like Alabama and Florida and Louisiana and Mississippi were created by Congress. They didn't join anything, they were admitted only after gaining the approval of a majority of the existing states through a vote in Congress. Since the other states had a say in allowing them to join in the first place why is it hard to believe that the approval of the other states was required to leave?