"One could argue that simply participating in the lotter shows that those kids had active parents, and therefore are self-selecting as the OP stated." Exactly

"One could argue that simply participating in the lotter shows that those kids had active parents, and therefore are self-selecting as the OP stated." Exactly the point: comparing students with equal motivation and parental involvement, who are separated only by luck, is a convenient way to control for the self-selection bias.