I don't think it's possible to win or partake in every race with a single chocobo. You should, in reality, focus on the races that allow you to progress up the rank or those that have specific treasures of value, such as the Dahaka stakes.
Either way, you can always retire your silver chocobo, renew it and try out the other races if you desire. --- For to win one hundred victories in one hundred battles is not the acme of skill. To subdue the enemy without fighting is the acme of skill. ~Sun Tzu