Many also have extra monetary incentives.
The best thing about a full ride is the the attention you are getting because you are in an elite group of students. You have a lot of opportunites that other kids in a public school sdon't get, even the kids in private school don't get.
I am not saying a rull-ride in a public school is better than a first tier private school. I am just saying that if it's a Coach, it's a special and expensive version of Coach.