Actually, with respect to most of the private schools that give the best financial aid, having divorced parents is disadvantageous for getting financial aid, since they require the financial information from both parents.
Meanwhile, most schools do not have good financial aid, so having more money is better. The almost-free need-based rides for students from lower income families are limited to those who can get into the few schools with the best financial aid, which are generally highly selective and do not admit many students from lower income families (usually under 25% Pell grant students).
Need-based financial aid offers at even the schools with the best financial aid typically assume that the student will contribute several thousand dollars from some combination of work earnings and/or federal direct loans.