I've just watched the Sunday morning live clip and I appreciate it's hard to discuss in a few minutes on mainstream tv. I'm glad it's getting this much oxygen and I hope that continues, but it still felt like only part of the debate. There was very little said that explains the difference between sex and gender, for example.
One point that does annoy me is the complaint that it shouldn't be this difficult to change your gender. But it's incredibly difficult to change any aspect of the data on your birth certificate, and rightly so, for the purposes of high standards in identification. If the documents we rely on for so many things are to have credibility, they must not be flexible.
I can't change my eye colour, or my birth weight, and to change my name requires application and consideration, review the deed poll process for more information.
Gender, being a social construct, is flexible, and therefore is less appropriate as a marker for identification anyway, whereas biological sex is what is recorded on the birth certificate precisely because it's inflexible.
Obligatory pre-emptying of the intersex issue, yes a small proportion of people aren't easily observable as one sex or the other at birth. We are still a sexually dimorphic species. In the same way that some famous comedians, Olympic athletes and other everyday people were born with only one foot, but that doesn't stop humans being essentially bipedal.