There is a lot of government involvement in American universities, but in weird pseudo-market ways, like grant programs (to institutions) and subsidized loans (to students). They're not a good example of the non-market model, which is much simpler: public universities, funded by tax money, staffed by civil servants, charging no tuition. Some U.S. states had this model through the 1970s, and it worked fine. Parts of Europe still have it, and it works fine, too.
I've been at both American and Danish universities, and the American universities today really do not seem progressive to me. Many professors will claim to be left-ish, sure, but structurally the universities are not run according to a traditional public-service model, and are not internally egalitarian. Their structure, with tons of administrators, large pay differences, odd systems of performance reviews, people job-hopping for pay raises, administrators desperately looking for revenue, etc., reminds me of American corporations more than it reminds me of public service. Just the pay inequality in itself is shocking, more like corporate levels than civil-service levels. An American Dean's salary might be twenty times or more what a PhD student is paid, and professors' salaries might differ by 2-4x even within just the tenure-track segment. In Denmark the entire range from "highest paid professor" to "lowest paid PhD student" is maybe 2.5x, and professors' salaries are on a boring civil-service scale with about 30% variation. This already makes the American environment much more like a company: competitive, hierarchical, not really egalitarian, big focus on revenues (for the institution) and salary/advancement (for the employee).
There are admittedly things the American university model does do well; it's well regarded in research, for example. But one thing it is not, is a straightforward public system along the lines of what a social-democratic government might run. (One might say the same about the healthcare system, for that matter.)