With horizontal stabilators, the structure passes through the fuselage and is structurally easier to support. A moving fin would just be a pivot with the flight control above it. It would take a lot of fuselage strengthening to prevent deformation and to support a whole moving fin, for a structure that would not get a great deal of use. A rudder on the end of the fin does a satisfactory job- a whole moving 'finilator' would be far heavier, and still end up weaker. I know of no aeroplane at all that uses this solution.