In all intends and purposes, they become the new sex they diagnosed with.
Who is qualified to make that diagnosis, though?
While I agree that some cases of gender dysphoria are completely genuine, I also believe that some are merely following a fashion/trend or doing it for attention. Others are somewhat unsure, as seen in some recent threads about transgendered people who later change their minds... one was as young as 12, I believe.
I further believe that, as better described by some of those doctors and psychologists, the dysphoria is due to the change (or lack of change) in society's understanding of gender roles and what it means nowadays to be either.
You can be a man and a nurse, for example, or a woman and a front line soldier.... some of the lines that previously used to define (in part, at least) your gender, and what it was to be that gender, have blurred... whereas in other walks of life, they have been further reinforced.
I have both acquaintances and colleagues who are trans, mostly female, and most of them admit they still (at least occasionally) question whether they made the right decision... almost as often and as casually, it seems, as I sit in rain-soaked traffic wondering whether I'd have been better off taking the bike.
But even if you don't transition, it still comes down to peer-acceptance. For some women, you just won't be female enough to represent them.
I think the question is a bit flawed here as we are making the assumption that gender is the most important factor for fair representation (why not class, for example). Are we trying to represent mothers or are we trying to represent women as a whole? Etc
You have to draw the line somewhere, otherwise you'd have every sub-group of every possible division screaming for equal and proportional representation, to the point where any suitably qualified candidate would need to be of a very specific gender, very exact racial mixed parentage, particular sexual orientation, social sub-class, graduating from certain type of school having studied specific subjects, having worked in certain jobs, lived in certain streets, supporting certain football teams, voting the right set of contestants out of the X-Factor, Big Brother and Love Island, who likes all the right music, eats the right food, etc etc etc........