One of the most misleading statistic that AO’s constantly throw out is the high percentage of applicants who can “do the work”. If we look at the most recent CDS of Yale, over 10% of the matriculated students had at least one SAT subscore in the 600’s. In fact, there were students with subscores in the 500’s. So by definition, the AO determined that those students “could do the work”.
As an alumni interviewer, I have had the opportunity to get periodic debriefs from both my Regional AO and senior AO’s about the admissions process. A consistent theme is the importance of the LoR’s. The AO’s are savvy to the fact that essays can be massaged by people other than the applicant, but the LoR is an independent third party evaluation of the applicant by someone who knows the applicant well and can compare the applicant to his/her peers.
For the Yale process, about 20,000 make the first cut with 6,000 going to Committee.
Looking at OP’s hypothetical, if we try to back in some numbers, this is what we see, the bottom 25% make up about 400 of the matriculated students and about 550 of the admitted class. I think we can safely assume that the yield on the bottom 25% is going to be much higher than that of the general admitees, so we are talking about let’s say 400-450 admitted student. There are approximately 200 athletic recruits, not all of whom will be in the bottom 25%, but a good chunk will, add to that development cases and other strong hooks (URM, first gen, low SES, geography), so we see the number of spaces dwindle pretty quickly for a non-hooked kid at the 25th or lower percentile. I am sure there are kids like that who get in, but they are going to be very rare. I also think when we talk about achievement spikes that AO’s will take note of, school level accolades will be a dime a dozen. I would think at a minimum, an applicant would need state level or perhaps city level (if we are talking a major metropolitan area) accolades for the AO’s to take enough note to overcome an otherwise below average application.
All of this having been said, if the hypothetical kid were mine, I’d throw the dice. The chances are 0 if you don’t apply. I’d be sure to have realistic low expectations on this though.