My Two cents on the matter...
If you feel like a woman, you are perceived as a woman, you fullfill your daily life as a woman... Why not? What it is under the legs, I can see it mores a medical issue. It's the same as a man that has been accidentally maimed down there, but he is still a man (unless you are using the Deuteronomy, 23:1). It is not something that can be changed that easily (not enough budget, or nor a safe surgery depending on your health) at will, and IMHO, not something that should affect you in your "social / public" life.
The process that they use in SP seems fine to me. After one year of HRT and therapy, if the therapist sees you are fitting into the new role, the change can be easily made, and they will try to change all the other related documents, as many as they can.
Yet, there is something that is bugging me as a consequence of this thread, and can be quite interesting...