First, you're right. Despite how it's described in the article (and it's described poorly, if not entirely inaccurately), this particular state statute isn't as crazy those passed in Tennessee and Oklahoma (OK might have actually been a Constitutional Amendment).
In any event, can you think of any private arbitration agreement that would be enforceable under existing US federal law, or South Carolina state law in instances where contractual obligations violate "constitutionally guaranteed right(s)"?
Neither can I.
Although, I would stipulate that this is South Carolina, so who knows what kind of crazy-ass provisions are secreted away in that state constitution. Of course, even with this state statute, any limitations or prohibitions on binding arbitration established by state law would (I believe) be superseded by the Federal Arbitration Act (see: Southland Corp. V. Keating, 465 U. S. 1).
Which of course begs the question, why. Why go through this ridiculous legislative circus, to prevent something that is already clearly prohibited by existing federal law and volumes of Supreme Court precedent?
After the Legislative body of South Carolina remedies the problems of Sharia Law being forced upon their frail citizens and whatnot, maybe they can then pass additional laws forbidding mad-scientist from creating earth swallowing vortexes, and stuff - 'cause you know, that's a big problem, too.