Schaefer's dichotomy theorem and reformulating 3-literal clauses
Schaefer's dichotomy theorem doesn't purport to claim anything about what transformations might be possible / not possible.

However, as Yuval says, we don't need Schaefer's theorem.  We already know that 3SAT is NP-complete.  Therefore, we know that if there is a polynomial-time transformation that transforms a 3SAT instance into an equisatisfiable 2SAT instance, then P = NP.  Put another way, if P $\ne$ NP (as many researchers suspect to be the case), then there is no polynomial-time transformation that transforms a 3SAT instance into an equisatisfiable 2SAT instance.  Put yet another way, finding such a transformation that provably works is at least as hard as proving that P = NP.

The same holds for a transformation to any other set of formulas that are known to have a polynomial-time algorithm for testing satisfiable (e.g., the conjunction of Horn clauses).