Then when it became career training, the government stepped in and provided a lot of money to help non-wealthy people enter the system, which was probably a good thing. The unintended consequence of that was that colleges starts sucking up all that money the government was giving out, creating a vicious cycle.
Government funded universities would solve this, because they would set the price, but we'd probably lose out on a lot of great programs. You could also solve this by outlawing charging tuition and switch college funding to being a percent of the graduate's future income. Of course, this would get the colleges to optimize for only majors that have high incomes after graduation, and we'd lose all of our philosophers and writers and artists (which I personally think would be a bad thing).
Maybe there is a hybrid approach here, were you can choose to pay now or pay later with a percent of your income. Those that choose pay later would most likely be forced to choose a "high potential earning" major, and those that paid now could choose whatever they want. But then this would lead us back to only wealthy kids being able to afford "the arts".