The banks would not go bankrupt. The government would be paying. They would not loose any students. The students would say "cool, I owe less money, thanks government" and then continue going to class.
I don't think it would be as simple as you're making it sound.
The economic impact of the cancelation of over $1.6 trillion would likely collapse the US economy. At the very least the economic impact would be felt for decades to come.
If Bernie decided to "cancel" the debt by just paying the banks $1.6 trillion in taxpayer monies, the banks wouldn't shut down; however, typically canceling a debt means it's just that, cancelled. The banks would ostensibly lose $1.6 trillion they had loaned out.
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