This is a really touch question with a lot of factors to consider.
Here is my opinion, without much fact to back it up. If I were king, here's how I would do post-secondary education, roughly:
- Private universities, non-profit and for-profit, can exist and mostly do as they please (subject to some accreditation requirements.)
- Set up public two-year colleges (CC), entirely funded by the government but mostly free to do as they please. Their funding is fixed and pays per student enrolled, with bonuses for each student who has successfully graduated, and further bonuses for their reporting either a job in their field, or a well paying job, or attendance of higher education. Actual definitions of those things are complex, just assume they would be reasonable and reasonably strict, to avoid things like graduating everyone and massive grade inflation. Extra funding for specific areas: lab equipment and other specialized requirements; one-time grants for construction and development; and so on. Offset for cost of living in certain areas (CC in silicon valley gets more per student than CC in bumblefuck, egypt.) Will admit anyone with acceptable grades automatically. Anyone not coming straight out of high school needs to apply, but with rolling admissions, it shouldn't be very difficult to get in: just a bit of a burden to show that the applicant is serious. However, poor grades in school result in suspension / expulsion, no need to waste money on people who won't learn. Standardized programs have standardized tests, to avoid grade inflation to keep students enrolled who shouldn't be.
- Set up public four-year universities, with post-bachelor's programs, again entirely funded by the government but mostly separate. Similar requirements for funding. Tough admissions requirements. Similar grades requirements. Pay top dollar for professors to attract top talent. Separate teaching from research: a professor may be hired exclusively for one, or the other, or both, but does not have to do both. Do everything possible to ensure high standards, high reputation, good education - this will attract top students, especially considering they need not pay tuition. Ensure that other student costs (rent, books, living expense grants) are disbursed liberally to any who need them. Have programs that attract top talent from outside the country, similarly at no cost, and ensure they have education and work visas.
Now every other private university has to compete with excellent free public universities. They can charge however much they like, but they'll actually have to charge what they're worth - today, every goddamn private university, regardless of how good it is, charges the same amount (roughly) because they can.
On the other hand, anyone who wants to attend a CC can also do so free of charge, provided they keep their grades up. They can focus on an academics path to further education elsewhere, or an employment path to learn skills that get them hired upon graduation.
High school is really not enough anymore, so if at least two more years are required for most people to have a decent career, then we should be paying for those two years. And yes, I will happily pay some extra tax to make it happen; even selfishly, more educated folks earn more money and buy more products from my employer, which keeps me paid as well.