Most psychiatrists/doctors in general do not really know much about stuttering to be honest. They are clueless unless they are speech pathologist and actually specialize in it. That being said, it is a protected disability at least in terms of discrimination. In terms of getting disability income it can vary. I believe you could get full disability if you make a case about how it is affecting your career drastically and your mental health. I know that ADHD can also qualify you for disability.