It wouldn't bother me at all. Ts are just people, too...and he seems to be self-aware and taking care of himself. I agree with your wife and see it as a positive thing. (But then I also find self-disclosure to be helpful in my own therapy.)
ETA: Are you concerned he won't be able to help you b/c of his own issues? Most Ts can keep their "stuff" out of the room with their clients, esp. if they are getting help on their own time.