I'm not going to let an employee with a mental disorder train new hires
That seems harsh. Why aren't you going to let him train?
He will lead by bad example. His mental disorder prevents him to work well with people.
If he is not up to the task or have failed to do his training duties in the past, then it is your right to select someone else to train the new hires. However, you shouldn't use his mental disorder as the reason why you would prevent him from training new hires as that can be a form of discrimination.
It was his job to train new hires and I think it is wrong for me to take it away from him.
I am glad that you agree. I do hope that your company succeeds with the new bunch of hires that he will train and I do hope he does well, and he shouldn't be defined by his mental disorder.