These robots are obviously designed to turn at that exact radius, and no other.
Not at all. The robots were designed with no specific turn radius in mind and can make much tighter turns. They're following the lines using PID, and the constants were picked through trial-and-error optimization on courses with turns no sharper than a 6" radius. To function well on sharper turns, it's possible the PID constants would have to be altered somewhat, or maybe the maximum speed would have to be decreased (on this course the robots can run at full speed without losing the line).
Now when my 3pi is running maze solver code it executes pre-defined 90 and 180 degree turns at the intersections, but in the above video the turns are not hard-coded. The robots don't have any idea that they're on a turn or a straightaway, it's all just PID.