Its already very common that such accommodation requests get filed with associated medical paperwork from a medical professional outlining why the accommodation is what the person needs. That alone should be more than enough[0]
Secondly, why don't we simply trust adults to make decisions about how to manage their conditions, especially if there is no demonstrable issue with how they work with their team and their work is up to standard.
[0]: Its been some time since I haven't simply produced such paperwork to go along with a request, but if you don't produce it upfront if I recall correctly employers reserve the right to ask for more information, which typically boils down to getting associated paperwork from a medical professional
There's abuse of every system. So should we just quit doing anything?
I don't think pet-food companies will be paying doctors to approve support animals, so probably most of this will have to come from the patients themselves.