Because the Republican party decided Lincoln was their man and Southern statesmen took that as an indication the empasse between them and the North had become unbridgeable. It's interesting that Lincoln's name wasn't even on the ballot until the third one which, when his name was vetted, only fueled states' decisions to secede. No, Lincoln didn't have any more of a place on the seceding states' ballots than Vincete Fox would have had on an American's ballot.
Lincoln was the legimate nominee for a major American party.
The Southern states kept his name off the ballot, not allowing the people to vote for him.
They were still voting for President of the United States and had not seceded.
Now, the fact that they found the GOP unacceptable was due to one reason, the GOP had a platform of limiting slavery.