The US is a ridiculously litigious country. It could end up being very, very expensive if they did their own assessments, even if they hired doctors to do so.

It is more litigious than the UK, but UK universities have Special Educational Needs specialists. It would be very very unusual for a family doctor (what we would call a General Practitioner) to be willing to make a diagnosis on this, they would insist on referring you to a specialist, although in many cases that may not be a medical doctor at all. ADHD, Nuerodiversity, Dyslexia are all assessed by specialists. In all professions it is considered unethical to act outside of your area of competence...

Admin staff or teachers of the university are not entitled to assess that.

That would mean the university hiring doctors, or at least paying a doctor to do the test.

They rely on the note in order not to do their own assessment. It's outsourcing.