I don't think there's any benefits to visitors (other than getting their fortune told)- people need to visit Katrina in their own town in order to unlock her shop.. Going to other towns to visit Katrina wouldn't help getting the 20 visits since she has to be in their town.
If you want to unlock her faster, as it was mentioned in sticky thread, make new character (new save file) in your town then visit her, and repeat the process multiple times.