I would not worry about a psychiatrist being bothered by an ex client being angry at them and I certainly don't think they want to do good or help people.
I also doubt hate mail bothers them much. But it might not be something that is good for one's own self to send. Such missives, particularly after the situation has ended, rarely work to cause the other to reflect upon their wrongful actions, probably won't get you a response, and won't change the mind of the person who quit you. In my experience of dealing with those guys as an attorney representing clients against them - they may place such letters/emails in the chart, consider them evidence of what ever they had diagnosed the client with, and go on with their lives.