(SPOILER)The are no times in the game were a Chocobo is absolutely neccassary.
False.

KoTR can be received without a Gold Chocobo.
True.

How do you make a Golden Chocobo?
Great + Great + Carob Nut + combined total of 4 or more race wins = 50% Green, 50% Blue
Green + Blue + Carob Nut + combined total of 9 or more race wins = 100% Black
Black + Wonderful + Zeio Nut + combined total of 12 or more race wins = 100% Gold

Are you allowing discussion about ambiguous answers?