Is that really the case? It seems to me that a Prolog-like resolution system, coupled with a constraint solver, would get most of the job done with little effort.
There are certainly many rules to keep track of, but in some cases the newer rules are strict generalisations of the older rules. For example, Haskell's original type classes are just a special-case of modern multi-parameter, flexible instances/contexts, etc. type classes. Likewise, we can implement constraint kinds, type class instance resolution and type equalities using one constraint solver.