IMO, the advantages of top privates is two fold: 1) they have more $ to throw at undergrads, so the chances are greater of getting involved in undergrad research; 2) recs from a well known faculty member at a top undergrad may be also be personally known by the top grad profs in the same discipline. (If nothing else, they attend the same professional conferences and cite each others’ work.) Thus, it is really easy to pickup the phone and inquire about a candidate.