I am not aware of any UAA league rules or agreements between the institution, although there may be some. For Chicago, coaches have less pull than, e.g., the NESCAC, so the recruiting lists tend to be long. I understand that coaches select their top two (depending on the sport) picks, who get likely letters. Additional recruits are chosen by admissions, based on their assessment of the application, from those remaining on the list. Not every athlete on the list is admitted.
For WashU, I understand that the academic credentials of recruits do not vary much from the non-recruited applicants.
I know of one supported athlete at Carnegie Mellon who was deferred and then rejected.
Your best bet is to have the conversation with the coach at the school of interest. I would first ask in a general manner how the recruiting process takes place. Then ask these follow up questions: 1) how many recruits on the list; 2) how many from the list are admitted; 3) how many are denied, deferred or waitlisted; 4) where the athlete is on the coach’s list of athletes; 5) how many athletes in a similar position to the recruit were admitted; 6) do you think the recruit will be admitted; 7) are there open tryouts for the team; 8) is the recruit is guaranteed a roster spot (at least for the first year).
Sometimes the pre-read doesn’t work out, especially if the sport only has a few spots. My DD was recruited and her pre-read was off the charts. The coach did not promise one of the spots as she waiting for a few others (better position athletes)…coach was honest and up-front. The other pre-reads came in positive so the coach communicated to my DD that she cannot support her application, but if she gets accepted, she can join the team. It turns out my daughter got in, however, the others did not. But that was pre-covid where the standardized test played a factor. My DD decided to go to a different school.