More likely hands-off parenting.
Two things about private schools that give them an edge (besides "recruiting"). First, when money is in the equation people have a greater incentive to get involve. Second, many private schools require their student to engage in organized sports throughout their time there. So private schools tend to have full roosters.
Public school parents don't have any skin in the game (beyond their property taxes). So there's less encouragement from parents to go out for sports, especially if there hasn't been any aptitude shown by middle school.
On the other hand, roosters are often full in rural schools because those schools represent the community their serve. It's not just a place they go to.