It's not that it doesn't work for less economical majors in less competitive schools. It's just that the market is crowded out by the availability of credit from government loans. If there were no government loans or were less restricted, banks would offer credit to those less credit worthy, however the interest rates would be higher (higher than those the government now offers).
So incoming freshmen might see options like this:
State school, engineering 5% interest
State school, lib arts 8% interest
Private school, engineering 7%
Private school, lib arts 12%
This would also be a huge signal in terms of assessing the value of a degree from a school. I'm not saying that those with higher interest rates are less desirable, but in terms of an investment, they are a riskier. The interest rates would be mostly determined by historical loan performance of those that went to those schools and majored in those fields.