Then why don't they offer all their programs for free? It would be very good pr and they would get many more donations.

Because they don't have to? Their goal isn't to create good PR, its to cement themselves as a rite of passage for the wealthy, intelligent, and influential. They don't really care what the plebs think about them, they just want to make sure that the select few coming from lower or middle class backgrounds but have the abilities necessary to move up in society are interested in attending their school. And those people are generally aware of the financial aid situation, or will at least apply and then find out about the financial aid situation if they're accepted.

Also, removing the tuition would make them look cheap to their target audience. In reality, a significant majority of students do end up getting financial aid. It just has to be indirect, so it doesn't harm the school's reputation.