Dana, happy birthday!!
I get the hurt of the "old man" remark. True, they don't know any different, but that doesn't make it hurt any less.
No, you are not old, either chronologically, or for starting your transition. The most common age to start transition is 35-50, and there are plenty of us over-60 folks around. You have many good years ahead of you, and most of them will be as your true gender.
Hang in there. It gets better.