Except that already is case. You need a degree by an accredited institution. The Department of Education regulates the accreditors, which audit the universities.

So we could have better or more aggressive regulation here, either at the DoE or the accreditors. Or even have the department accredit directly, but I suspect these institutions will still likely game the system.

Usually the bargin-basement universities do actually have worthwhile material, the poor ones just don't provide you with the full support to succeed, and often overstate the value of their program relative to alternatives (or cost). I mean, you could just buy textbooks on Amazon and do self-study. And for some folks, that might even be a good program! Good universities provide a lot more than just the lectures and textbooks, but I suspect setting standards to assess those properties is difficult. Even going to a well-respected state school, I had a few classes that were very poorly run. In one case, my dorm-mates and I created a study-group and basically taught ourselves the material. But I shudder to think of a whole university consisting of classes like that, though I suspect they would meet baseline audits of the curriculum. An alternative approach would be to assess the outcomes of students, but looking at the can worms that raises in primary eduction, I don't think that is a good approach either.

Pray tell how would one prevent gaming of the auditing.

As an aside, my dad worked at the VA. There was a person who was supposed to be promoted to CIO and then they did an audit and found her undergraduate credentials were not correct (don't remember what the exact circumstances were). As a result she got canned instead of promoted.

There are positions in the federal government system where you are required to have a college degree or else you are not eligible, no matter how competent you are.