To prove that a problem is NP complete you have to reduce SAT to that problem, not that problem to SAT. You can solve combinatorial search problems with whatev

To prove that a problem is NP complete you have to reduce SAT to that problem, not that problem to SAT. You can solve combinatorial search problems with whatever method you want, but it can be very hard to beat a SAT solver, because you may have to duplicate all the strategies, engineering, and heuristics that SAT solvers have. If your problem has special structure then it may be worth writing your own solver. Constraint satisfaction problems can often be written with high level constraints that have good constraint propagators (e.g. the alldifferent constraint).