1, entire sets could be built around the countering, or counter countering, of that one move.
2, Ice types would still be the only Pokemon that could hit Dragons hard, and they are fragile, and though water types can sub, they don't have stab
3, the fairy type made poison types and steel types more powerful as well.
4, the fairy type can be dual typed, and has one of the smallest types in Pokemon. Furthermore, there is more then one fairy move, and it was added with the mega evolution. All of these variables at once caused so many changes no one could be a problem, where one move added with a specific purpose can cause chaos.
5, fairy types are unique, weak to two usable yet neglected types, poison and steel, immune to dragon altogether, and a space attacking type weaker then psychic, yet strong then ice. adding (though many including myself were reluctant to admit it) much needed diversity.
6, and duel fairies such as Mega-Altaria, and Gardavior are some of my new favorites.
Some the Fairy type was the system update Pokemon needed, rather then the patch one move would have been.
So, No, it wouldn't work. However the idea itself was rather clever, and a valid point, in fact in gen 4 it might have been an option.
Hope I helped