I'm not convinced you would need to. "Students" cease to be students immediately after graduation, so they wouldn't really be "students" anymore, right? So "none of the downsides to bankruptcy really apply to students" doesn't really seem accurate, does it?
Even reading it more charitably, students and recent graduates still probably would like access to credit cards, or the housing rental market, or whatever. Bankruptcies stick with you for 7(?) years, so you'd also have to think about whether you'd be locked out of the mortgage market, auto loan market, what-have-you, while your peers are able to make those moves.
Bankruptcies are also like a whole legal thing with a judge and everything, so I doubt we'd see every single student getting their whole debt discharged instead of a judge just being like "Didn't you, like, just spend all this money on getting the training required to get a high-paying job? Seems like if you plan on working anytime soon you should make some payments."
Also lenders can just like, be more selective or restrictive in other ways? Higher interest rates, requiring more established co-signers, etc etc etc. Honestly the strangest part of this is that we've normalized saddling 18 year-olds with tens or hundreds of thousands of dollars of debt.
Bankruptcy stays on your credit report for 7 years, but doesn't lock you out of the credit market. Someone with a high paying job has relatively easy access to credit regardless of recent bankruptcies. At most it effects them for a year or two, but new graduates aren't really making any big purchases using credit immediately after graduation anyway.