This is political correctness to the extreme, particularly for departments in the science field such as geosciences and molecular biology.
The duty of graduate schools, particular for Ph.Ds, is to teach people to further human knowledge based on existing learning, and GRE scores are an indication of how well one has absorbed fundamental knowledge in a given field. Therefore, the advertised approach is not recommended.
The given reason that the GRE test is too expensive is just an excuse. If that is the only reason for trying this less than thoughtful idea, particularly for departments in the science field, why not provide some funding for students to take GRE test in schools with a high representation of underprivileged minorities? Surprisingly, such a simple idea is not tested first to 1,000 “worthy” students, which surely is much less expensive then providing one year's funding for one student in each of the 14 “enlightened” departments. I feel sad to support Princeton so that my beloved University can wantonly throw my hard-earned money away without doing any basic test first.