"planned major" role in admission to elite schools?

It may be helpful if you tell us the schools. But, in general, the answer is no. If it’s a university that has different undergraduate schools or colleges (Cornell, NYU, Northwestern, Michigan, e.g.) of course you have to apply to the specific school. Same with schools that just have liberal arts and engineering. Have to specify liberal arts or engineering (e.g., Tufts, Lehigh, Union). But if it’s Yale (non-engineering) or Williams or Middlebury, no, it does not matter if you specify that you intend to be a psychology, biology, or philosophy major. They know that most students change majors. Many do more than once. That said, maaaybe if the student says he intends to major in classics and has substantial background already in Greek and Latin.