With very rare exceptions, BS kids cannot compete in club sports. The primary obligations are to the school and its required activities which make it difficult/impossible for even a day student to leave the campus before 6. Combined with the fact that many BS have Saturday classes, the logistics become a challenge.
So while BS kids do get recruited, particularly in more niche sports, the 1% of the 1% high caliber athlete is usually competing at the club level and attending an LPS/day school, specialized sports academy, or is homeschooled.