Congrats on getting all the dragons!
I think LSnake's advice is to have lots of activity going (have dragons breeding in both caves, building shrines, habitats, etc. ) before activating the special ability of Kairos. His tips was assuming you already have Kairos not for getting one. Once you activate Kairos' ability all your task will jump 6 hours forward thus will be done faster.
You can name some of the dragons you mostly use for the races temporarily for now then rename them when you got Kairos. You can get fragments on both the Prech and the race track. I got the 2nd to the last piece from the Race Track and I've gotten many pieces from the Perch as well. One tip I would give is try not to get a GOOD on the race. Preferably get PERFECTs and GREATs. Good luck
GCID: *.Philip35.*