This is the loophole. Universities aren't the ones diagnosing, they're the ones accommodating.
The current meta-game is for parents and students to share notes about which doctors will diagnose easily. Between word of mouth and searches on Reddit, it's not that hard to find doctors in any metro area who will provide diagnoses and accommodation request letters to anyone who makes an appointment and asks nicely.
There are now also online telehealth services that don't hide the fact that this is one of their services. You pay their (cash only, please) fee and they'll make sure you get your letter. The same thing is happening with "emotional support animal" letters.
Once it becomes widely known that getting a diagnosis is the meta-game to getting housing priority, nicer rooms, extra time on tests, and other benefits the numbers climb rapidly. When the number is approaching 38%, the system has become broken.
It's a real problem for the students who really need these accommodations. When 38% of the students qualify for "priority" housing, you're still in competition with 1/3 of the student body for those limited resources.