Are there any legitimate reasons a university may have for NOT
providing needed accommodations for students with documented mental
Legitimate reasons, no, absolutely not.
Should universities give extra time or related accommodations for
students with mental illnesses?
A student with a handicapping condition should look at lists of possible accommodations, and do plenty of experimenting, to find out what's helpful. The university has a moral obligation, and in many places, a legal obligation, to provide needed accommodations. Not to do so constitutes discrimination. In the United States, a refusal to provide needed accommodations is a violation of the student's civil rights. An educational institution that refuses, risks losing all federal funding.
The nature of the handicapping condition is only relevant to the choice of accommodations, not to the initial decision of whether to provide accommodations.
Here is one list of possible accommodations -- there are a number of them on the web. Reading them is a great way to brainstorm what might fit best for a particular handicapping condition and for a particular student.
I will provide one small example. My son sometimes, but not always, writes and erases each letter, or word, or sentence, sometimes multiple times for each such unit. At first, this symptom occurred only with handwritten work, but then it crept into typewritten work as well. The solution: a scribe. He dictates, the teacher types; she reads it back and shows it to him, and he can edit the text himself, or dictate edits. This accommodation is very helpful for him.