I appreciate the exposition but I think one key thing is important: the base argument only relies on conditional independence, and it is likely that this is not violated. In particular since Harvard and Yale are unlikely to collude openly then it is probably true that P(A|B,C)=P(A|C) and P(B|A,C)=P(B|C) where C is personal characteristics. Your current argument suggests that the correlation is passing through those personal charactersitics but since those are constant it makes it likely that the outcome is conditionally independent.
If so (and I think it is, I find it hard to believe that college admissions are inherently dependent since they do not share staff)— the complicated bayesian updating is unnecessary and you just need to select an independent probability for each tier.
Great idea to add some stats to this generally superstitious activity.