I'm a huge fan. Don't always agree with Dr Greer's views, but she is a pioneer whom we dismiss to our own detriment. People should read The Female Eunuch - it's revolutionary.
On this morning's Today programme, she simply said that she had nothing against peole transitioning from one gender to another, but that to say that men became women was nonsense. And it is at the level of sex (as opposed to gender). She's right - apart from physiologically intersex people, most of us are largely (not completely) biologically one or the other sex.
But gender - the social construction of identity and power hierarchies consequent on that - derived from ideologically-formed views of what is appropriate identity for each biological sex - Gender is something else.
Over all of this, I can't help feeling that if we stopped seeing gender roles as fixed (femininity, masculinuity etc) then the issue wouldn't arise. If a biological man prefers to express his identity via traditionally "feminine" behaviours, activities, dress etc - well, really what IS the problem?
Men in dresses? Meh. Men who have sex with men? Completely normal. Men who wear make up? Normal in the 18th century so why a problem now? Aggressive men demanding access to women-only spaces: get off my patch.