The US can look at how other countries, that don’t outright provide free education, do it instead of reinventing the wheel.

Getting rid of the discharge protection is only a small part of it.

It’s more important to set a legal maximum for college tuition for accredited institutions.

Then make any subsidies and funds contingent being accredited.

Lastly make federal loans contingent on enrollment to accredited institutions, with the additional benefit of being able to cap the loan amount at a level correlated to the legal maximum tuition (not to be confused with setting at the tuition level because living expenses need to be taken into account as well).

Make the interest rate sub 1%, because the government shouldn’t profit off of you as it is a service and do away with private middle men that administer the loans, instead establishing a special loan administration agency.

This will have as effect that institutions either get in line or lose all government funds and a significant portion of enrollments.

If you then also manage to uphold a uniform quality level that you regularly inspect at the accredited institutions, you’ll end up with a clear, affordable choice of quality education v. unknown quality education that may or may not be a huge waste of non-publicly provided money.

ETA:

You can even take it a step further and follow more examples from abroad in terms of acceptance.
Where you aim to get to a situation that everyone that applies with the pre-requisite prior education credentials, gets accepted.

The way this is often done abroad is with a centralized application process managed by the government, in which you indicate your top 3 preferred colleges, the portal verifies your prior education (as it's centrally registered) and then enrolls you in order of preference.
For some studies, like law school, med school and psychology they'll have more applicants than available spots, and in those cases it's decided by lottery with slightly weighted chances based on your grade average.
The end result is that the vast majority of people automatically get accepted and the ones that don't get in via the lottery are almost guaranteed to be placed the following year.

This solves the whole minority/legacy/etc. acceptance debacle, makes applying for schools less like applying for a job with writing essays and stuffing your resume with a bunch of extracurriculars and in the process makes the accredited institutions even more attractive compared to the potential hold outs that keep doing things the old fashioned way.