I recently went back to school and got a look first hand how LLM's are used in classrooms.
1. During final exams, directly in front of professors: Check
2. During group projects, with potentially unaligned team-members: Check
3. By professors using "detection" selectively to target students based on prohibited groubds: Check
4. By professors for marking and feedback: Check.
And so, the problem is clearly the institutions. Because none of these are real problems unless you stopped giving a shit.
Good luck when you point out that your marked feedback is a hallucination and the professor targets you for pointing that out.