The primary obstacle for young adults seeking to complete a college degree isn't that their public schools failed to prepare them or that their colleges somehow alienated them to the point of dropping out. It's money. Even solidly middle-class families can seldom cough up the more than $160,000 that private college will cost over four years.Virtually all private institutions (certainly those of quality) offer need-based tuition waivers and partial tuition waivers. Many private institutions offer "no-loan" packages to poor families. As a rule, the lower family income, the more generous the funding package. Of course, if any price is too high a price, the author can be guaranteed that truly deserving students are offered merit-based awards.
In addition, colleges, especially private ones, live off of endowments. These institutions understand that funding poor students for four years will likely payoff in terms of future donations to university from these students. In fact, poorer students' lives will have improved relatively more due to their post-secondary education, and we should expect them to be relatively more "grateful" to their undergraduate institution--especially if tuition was free. In this sense, colleges are motivated already to fund lower income students, and certainly are in a better position than the government to select high-quality low income students for need-based aid.
Finally, increasing aid may make schooling less worthwhile for everyone. One purpose of schooling, as Michael Spence has argued in the Quarterly Journal of Economics is to provide a signal to employers. The more bright a student is, the easier school is for them on every level. School becomes not worth the effort for the worst students first, and so the fact that someone achieved a degree provides a valuable quality signal to employers. Certainly taking away the cost component of schooling will differentially impact rich and poor students. Wealthy students will make a drop-out decision based on a far more lucrative outside option, and so we should expect a relatively weaker quality signal from poor students who graduate. Of course, this decreases the very "value" of education on which funding would be based.