Your sex cannot be 're-assigned' and whilst socialisation and personal character certainly make a certain amount of difference to one's self expression and modes of behaviour - it doesn't change one's inherent sex based blue-print.
I would say that this is not the whole picture for that tiny cohort (maybe <1 in 10,000) that have a mutation partially vitiating testosterone response. It's not obvious at birth what gender identity will develop, and as many as half develop a strong female gender identity during childhood, even if raised as boys and despite being karyotypically and gonadally male. (Cf individuals with complete androgen insensitivity - also karyotypically and gonadally male - in whom development of a male gender identity is virtually unknown.)
It ought to be obvious that infants' genitals as a matter of principle should not be surgically altered when they can't give consent, unless necessary to preserve health, urinary function etc. But there's also a practical reason, which is that the child must be involved in the decision later on, whether to have feminising or virilising treatment (or no treatment): they have a choice to make.The subjects of the C4 programme were examples of the wrong choice having been made on their behalf, when a wait-and-see approach should have been adopted.
And, their sex can be reassigned, in the sense that, if they are registered male, but choose feminisation, or vice versa, they can correct their birth registration with retroactive effect.
None of the above is relevant to trans people. Even if they have a sex-incongruent gender identity (about which I must remain agnostic), it's of no therapeutic relevance. They have normal genitals, which match their gonads, so they don't need any treatment for which gender identity would be a useful aid to decision-making.