Immediately yes, fortunately, there is a supply and demand curve, so if you cut the demand for a bunch of colleges that already exist; then the price should find a new equilibrium. Of course, as is the science of economics, time and outside influences matter.
Fortunately, in a low unemployment world, also means that hiring should be very elastic to this change.
You could apply that argument to anything. "Good luck with that, now people can't afford mansions, Good luck with that now people can't afford private jets".
Should everybody be able to afford mansions or private jets? No.
College should either be free, thus directly paid for by the government, or not everybody should go to college, you can't have it both ways, or that's how you'll get your big next financial crisis. The government should not back private loans, period.
Cost of education has skyrocketed BECAUSE banks started to loan crazy amounts of money to every students. This is free money for the banks.
It's a good thing they don't, then.
People have a different emotional response to predatory lending like payday loans, but not those same practices in student loans.
To be clear, I agree with you, rip out the loan guarantees and let the private market deal. But progressives would never allow this.