Compare the aid awards, and then make the decision as a family. Better to explore all options than to regret what might have been. It is true that financial aid from the federal government is largely limited to loans and a job on campus for most middle -class students who qualify. Unless your daughter has a 90-mph fast ball, it's also true that public universities give primarily federal aid. At private colleges, though, federal aid is typically only a small part of total aid available.
Private colleges use money from their endowment to fund grants.
Grants are outright gifts of money based on your financial need. Since grants are based on need, many students are eligible to varying extents. The more a school wants your student, all other things being equal, the more generous the aid package will be. Money will likely follow. The best school for your student is the most selective.