I can feel the frustration in your post, and I’m so sorry you aren’t finding the disability services coordinator helpful. I would encourage your student to find out who this person reports to, and work your way up the ladder. They likely report to a Director of the office or, if it’s a small school and they work alone, they report to a dean of some sort. You could certainly file a complaint with the Office of Civil Rights, and the school will most likely get into trouble for their process, if nothing else. That won’t necessarily answer your question though of whether or not he can have a course substitution. They will require the school to follow a process to get the question answered, but that’s it. They won’t make an accommodation determination.
When you find a supervisor to talk to, your son should be prepared with (1) his diagnosis, (2) his symptoms, and (3) the barriers he encounters in this class as they relate to his symptoms. He should also be prepared with a request of some sort. He may want to ask for an accommodation that allows him to give his speeches one-on-one to the professor OR that allows him to pre-record himself and then play it for the class. You may have more luck with accommodations than with requesting a course substitution.
If your son is certain the substitution is the only option that will work, the school must consider the request on a case-by case basis. According to OCR, determinations about whether or not a requirement is essential to a degree program or course must be determined on an individual basis by a team of qualified experts (not by an individual faculty member of dean). If your son is denied on this basis, he should insist on being provided a written explanation of the process that was used to make this determination. The key here is to be sure there is a proper consideration given to each request.
I find it hard to believe that public speaking is essential to every degree at the school, but I do believe it can be appropriately considered essential to certain degree programs, such as business, political science, and even education. If they determine that public speaking is essential to the degree program at their institution, and they’ve used a proper process, it is very hard to argue that a student shouldn’t have to meet the requirement. This is a core difference between K-12 and college. And that’s where the accommodations come in.