The only issue I have with making different colored chocobos have different abilities is that you get into the whole color=stats thing we had with gear that nobody liked. If you bred for abilities/colors separately it would be great.
Different functionality != different stats. In other games, e.g. FFT A2, different chocobo colors had different functions, almost different jobs even. It's more like customizing the automaton in FFXI- e.g. the black chocobos would fly, yellow ones had a healing ability, red ones had a fire attack, etc. They're distinct enough that each one has value. If it was just stats I would agree, but different breeds of chocobos did different things in previous games. There's a precedent for that.