It is certainly true men do not have to be the biological parent to be recorded as the father. It is certainly not true that the mother can put anyone she wants.
However, this thread is about whether someone who is the biological mother can be recorded as something else.
I think it should be clear, for the sake of the child, who their mother is.
It may be possible to make this clear and still to record this person as legally male.
However, as others have pointed out, it does rather show that the ‘living as a man/woman’ bits of the GRA are, well, vague in their legal meaning.