The problem here is that the patient doesn't have a general practitioner (GP, I will call it your/my "doctor" in the rest of the post) or that apparently there's huge wait lists. If I call my doctor, I can get an appointment within 2 work days or roughly half a week. And everyone in my country (The Netherlands) is insured for such visits. In fact, if I want to see a psychologist or physiotherapist or psychiatrist I first need to see my doctor for a reference, for my insurance. That's how it works here. Your doctor is your entry level. So, why not go to your doctor if you have a health issue? Here you would, but apparently in the country this person is from (I suppose USA) that's nigh impossible.
Also, employees have medical privacy here. Yes, a doctor could write such a letter, but ultimately the patient has privacy. Many doctors here flat out refuse such written statements, saying it isn't necessary. Though that might be on a case by case scenario.
On the other hand, I do understand the employer's PoV as well. If person X wants something special, then person A, B, and C also want that something special and before you know it you have half of your employees with -say- all kind of odd food allergies. Although it seems in this case the employee brings their own chair; here that'd be unheard, the employer would arrange a chair for the employee which suits their special needs. So, I disagree that this is a bullshit procedure but I am unsure how to solve the issue in a way where privacy is guaranteed, and then I say: take the loss of privacy in this case for granted and get the written sign.