The schools would undoubtedly respond that in doing so, they would necessarily have to stop educating people from anything less than the most privileged (and thus low-risk) backgrounds.
You'd need to cook up some sort of risk-pooling scheme if you wanted schools to extend that credit to anyone coming from a background that made payment statistically less-likely. And once you started doing that you'd have to be careful not to distort the market so much that the original intent -- to allow market forces to have more say on what majors college students go into, and who (and how many people) colleges admit -- would be totally lost.
It's a difficult problem and I don't know if it's better to have the schools/colleges trying to underwrite the loans. It might be better to leave that to specialists rather than force it onto schools (though forcing schools to release the data that an underwriter would need to rationally price loans for students to go there would be good).
If you didn't have the government backstopping student loans, and in some cases making them non-dischargeable in bankruptcy, that would probably have much the same effect. The government backing and unique legal treatment are what makes a "student loan" different than a giant personal loan. The justification has always been that without this treatment and guarantee, nobody would make a $100k+ personal loan to an 18-year-old, which is probably true. But I think we're starting to see the other side of the coin, which is that maybe there are some good reasons why that's the case.