Why would David Reimer ever have accepted that he was female simply because he was socialised as a girl?
(I actually don't think he was really socialised as a girl, but for the sake of argument let's pretend he was).
If I was told my entire life that I was a girl, and then I got to puberty and very obviously wasn't, because I had no uterus, no vagina, no normal breast development, no female hip structure and so on, why would I continue to believe a lie?
It is rather like if someone told you over and over again during childhood that you couldn't walk, and then one day you found out you actually could, why would you keep maintaining the lie, simply because you had been socialised to believe it?