OHMom is correct that lots of not-so-selective schools use big sports programs to attract full pay enrollees, especially male enrollees. Often referred to as “heads in beds.” Lacrosse is great for that. If a school already has football or soccer in the fall, it costs very little to use those facilities for lacrosse in the spring. Many D3 schools will have 150-200 guys on the football roster.
But those school are not typically going to be need-blind/meet-full-need admissions type of schools (which is this thread topic).
I’d agree that Harvard and Williams didn’t build up large sports programs for the sole purpose of attracting more full pay students. They have big sports programs because they like having big sports programs. Similarly, they don’t maintain sky high admissions standards for the sole purpose of attracting more full payors. But sports and high admission standards do have that effect in practice.
Legacy admissions is one that has a much more direct linkage to attracting to full payors. A binding early decision program is another one that pretty directly links to increasing full payors.