Perhaps this is anachronistic, but looking at the current operation of the US government it seems that a committed political group, even one that is not necessarily a majority can block substantial political changes.
With this in mind why was the southern planter class so frightened by Lincoln's election? The democrats still held a (narrow) majority in the senate and Taney was still the Chief Justice.
Why was succession seen as necessary when obstruction could easily have punted things down the road for decades?
There is a great book called “The Road to Disunion Volume II” by William Freehling which covers this topic.
Despite losing the House and the Presidency, The South still had the votes in the Senate required to stop any major problems. As you wrote, the Constitution still protected slavery where it already existed and the Supreme Court could be trusted to maintain that in the court system. So, many pro-Union Southerners argued to stay awhile and only secede if Lincoln actually does something.
Secessionists had the following reasons to push for secession immediately:
The agitation of a “Black Republican” winning the election ignited a lot of passions in the South and it was best to secede immediately rather than let Lincoln have time and then those passions dissipate
Even if it was true that Lincoln wouldn’t touch slavery where it already existed, South Carolina and other established slave states demanded the ability to expand slavery into new territories. Lincoln absolutely was refusing to budge on that topic as that was the core platform of his entire election campaign.
The Southerners feared that Lincoln would spread abolition thought into the South. Much of the book covered the fear that the banned book by Hinton Helper would incite “poor white trash” to revolt against the plantation class. Lincoln could use patronage to set up the Republican Party in the South and open up the mail system to allow such freedom of speech and thought around abolition to spread into the South.