Actually, it would.
The problem comes from bankruptcy laws. Banks were having a fit because students would get loans, and when they graduate, declare bankruptcy and have the loans forgiven. The showboat case for this was people graduating as medical doctors, declaring bankruptcy, then getting a high paying job. Banks went to the government to 'fix' this problem and the fix was: Student loans are immune to bankruptcy. If you get a student loan, you will pay it back, even if that means you will have your McPaycheck garnished.
This now means that banks have little to no risk to their loans. Why would they refuse any loan? If the student is successful or not, either way, they get paid. A kid that gets straight D's in school and wants to major in "Classical Nintendo" Sure! Here's money with a nice interest rate.
Now we have banks giving money anybody who wants it, demand for higher education goes up. When demand goes up and supply stays the same... prices go up. Prices go up? Get a loan!
It's a self-feeding model that all started with crony-capitalism. Banks and Government got in bed together. Now I don't blame the banks for complaining... it is a problem. If I lent off a ton of money for students to become doctors, and they kept stiffing me, I'd be pissed. But the Government gave the Banks too sweet of a deal. They gave them a win-win.
What if the banks had a 10 year probation window on student loan bankruptcies instead? If a student declares bankruptcy, the loan is put on hold for 10 years with no interest. If during that 10 year time, the student finds a job that could may payments, the loan sticks. If they can't after 10 years, the loan is forgiven. Banks are protected from those "evil doctors" getting hefty loans then dumping them... Students are protected from not being able to find a job afterwards. The loan (and cost of the education) must reflect the job that is received in the end. Students with poor grades, and majors that aren't in demand are less likely to get loans, as they are now risky to the bank. All of this should lower costs of education.