If the school gives four students $5,000 scholarships on a $20,000 bill - the school makes $60,000 and the students feel good about themselves. If they give one student $20,000, they make zero. At the end of the day, someone has to foot the bill - and if it's not the taxpayer, it'll be the people who can afford to pay.
Of course, high quality education should be available to everyone, but as a society we have to be more lucid about where the money is coming from. If taxpayers want people from low income families to go to school (and I am firmly in this camp), taxpayers need to be willing to pay for these people to go to school.