SAT, SMT, and constraint solvers are criminally underutilized in the software industry. We need more education about what they are, how they work, and what sorts of problems they can solve.
In what way? They're useful for toy problems like this but they're very slow on larger problems.
SAT solvers are used daily to generate solutions for problems that have literally millions of variables. So, what you said is just wrong on the face. Yes, some talented people can write custom code that solves specific problems faster than a general purpose solver, particularly for easy special cases of the general problem, but most of the time that results in the programmer recreating the guts of a solver customized to a specific problem. There’s sort of a corollary of Greenspun’s Tenth Rule that every sufficiently complicated program also contains an ad hoc, informally-specified, bug-ridden, slow-implementation of half of a SAT or SMT solver.