I don't think anyone has and "legit" info because each school's policy differs from the others and can change over time as well. And quite frankly, I don't think admissions wants us parents to know that they use a dart board to determine admissions because someone might include a magnetic application to attract darts (jk). Really, we parents all want to game the system, so the schools try hard not to give us any real information.
What I can tell you is that every school asks coaches for lists of recruits ranked top to bottom and most (but far from all) ask how many players you "really" need this year (or risk not having a team in certain cases).
At the larger highly selective schools, being at or near the top of the coach's list almost guarantees admission for a student falling into the mainstream of accepted applicants (above the bottom quartile) as they want successful teams and don't need ever student to raise the average score. It may also get you a more favorable FA read from the better endowed schools.
Occasionally, you will see schools where certain coaches are on the admissions staff. These coaches don't need to do as much arm twisting to get what they need for their team. Often their teams are very competitive. Some coincidence???
In our personal experience, the coaches at these schools don't seem to claim that they have a particular pull with admissions, although they do admit to sending in a list of preferred recruits.
I think that schools don't act alike on this matter. Different schools, different priorities. Some may look for full-pay, some (especially smaller) schools may look for multi-sport (multi coach recomendations) types over single-trick ponies.
And of course the sport does matter. Different sports are more important at different schools. In addition, one basketball player, or one ice hockey player can have a bigger impact on a school's athletic profile than one football player or track team member, as these are smaller teams where one player does make a bigger impact. Exceptions apply for skill players in football, though when the player is D1 college material.
Bottom line is that it is highly unpredictable because schools vary and they don't want you to game them (despite our best efforts
Sorry if I cannot be more helpful.
Holy cross-post Batman! I think we are thinking alike here!