I say to always have them on S class because stats carry over. It does make the breeding easier (as ive breeded A classes and just got a yellow chocobo) And if you breed S classes the chocobos become easier to control (I have gotten a black with B class green and blue, and he was a pain in the butt). Plus if you want to complete the chocobo sidequest and win the 10 races in S class with a Gold you have to beat that hard guy (Joe??) and you get the *SPOILER*Sprint Shoes, Cat's Bell, Chocobracelet, Precious Watch, and Counter Attack.