That as "at not cost to the student." Yes, society as a whole has to pay for education, but the long-term benefits more than offset that cost. Better-educated people are less likely to fall for scams, they are more likely to make good decisions about their health, etc. Democratic systems require educated populations: poorly educated voters will make really bad decisions about who or what to vote for, or will base their votes on things that are utterly irrelevant to society's problems.
This is not terribly different from the government paying for scientific research (disclosure: I am a grad student who is paid out of a grant from DARPA). Government-funded is a key component in scientific progress; while industry-funded research can make strides, there are certain lines of work that do not have a clear enough market potential for any business to justify funding, but which have expanded human knowledge and in some cases opened new markets (and thus paved the way for further research by the private sector). The public as a whole benefits from government-funded research in the long-term, even if it means higher taxes.
So while I will admit that you are correct in a technical sense -- education does cost someone, somewhere -- the bigger issue here is not whether education will be paid for, but who will benefit from it once it is paid for. Our current system of student loans has the effect of ensuring that people from a certain minimum economic background are more likely to benefit from education than others, and the wealthier a person's family is the more able they are to benefit from a good education. The effects are not hard to see: poor people are more likely to make bad decisions about their health, more likely to fall victim to scams (including but not limited to the lottery), more likely to commit crimes, and are less able to improve their lives or organize themselves for change. It would help to improve high school education, but that can only go so far.
The real issue here is the attitude Americans (and to be fair, plenty of other country's citizens) have about education. People tend to assume that college is high-end vocational training, where people go in order to get a "good job," which is typically defined as a job that pays a lot of money. The result of the popularity of that view is what we see today: a system of student debt, which has undermined the quality of college education and somewhat ironically undermined the confidence employers have in job candidates who present a college degree. With students who only care about the amount of money they will make after they graduate, who often try to rush through school so they can accumulate less debt and can begin paying it back faster, colleges have relaxed their requirements, slashed the budgets of departments that lack a clear vocational mission, and pressured professors to not make courses too rigorous. You see this sort of thing in computer science, where many departments only offer one or two theoretical courses per year, and where students can graduate having little to no understanding of fundamental CS questions (like the P vs. NP question) or who cannot state the answer to the Entscheidungsproblem or explain its significance.
Were we to adopt a different view -- the view that a well-educated populace is necessary for a healthy democracy and that society needs a critical mass of people who are capable of thinking for themselves as a protection against the formation of an aristocracy -- we would never have wound up in this situation. There would be no question that education should be paid for as a public good, certainly not in a country whose constitution forbids titles of nobility and guarantees that all citizens have certain rights regardless of their place in society. In theory, that is why we have a public education system at all; in practice, our public education system has been systematically undermined, and the voting public has long forgotten why we have public schools at all. At the rate we are going, there will be no public education at all when my hypothetical grandchildren are born, there will only be vocational academies and private schools, and poor people will only receive the minimum training needed to work in service and maintenance jobs.