It was more complicated than that.

Slavery was a major factor because the southern economy depended on it. Lincoln did not campaign to end slavery, he did propose to limit its expansion to new states. An amendment to end slavery would have been hard to pass, had the southern states not seceded (requiring 75% of states to vote in favor of such an amendment). But the loss of power in Washington could be devastating.

The “economic and political” interests of southern states differed greatly with the rest of the country because of slavery the southern economy was based on cash crops for export.

With Lincoln’s declaration that no future slave states would be added, it became clear that slave states would have a declining influence in the national politics of the nation.

That meant future wars or embargoes against countries could destroy the southern economy even if such actions were in the interests of the country as a whole. In essence the Southern states saw membership in the US going forward as a negative.

So the decision to secede was based less on the fear Lincoln would end slavery and more on the fact that the economic and political interests of southern states would be subordinated to those of the rest of the US.

Up until the Civil War the question over whether state membership in the US was optional like in the EU today or permanent had never been seriously tested.