As a D.O. She is still a doctor. Training at MD and DO schools is just about identical except that the OMM instruction is additionally done at DO schools.
But back to the OPs question. Private vs public doesn’t matter one bit. The medical schools are going to screen using your GPA, sGPA and MCAT score. If you make that screening cut, you might be invited to an interview.
My opinion, save as much money on undergrad school as possible, because medical school aid is loans, loans and more loans in most cases.