What a strange third option to give.
When it comes to gender, I think anybody should be able to live the life they want freely and without fear of discrimination. And if you knew me, you'd know I couldn't care less what people are whether referring to gender, religion, whatever, as long as they're not doing anything illegal or very obviously immoral.
However what frustrates me slightly is something I notice creeping in here and there, and it relates to masculinity and femininity. Let me give you a low level example. I was at mum's and she had a shopping channel on. They were flogging hand and body cleaning lotions. There was a brand ambassador on the show and the main presenter, both female. When selling one of the lotions, the brand person said along the lines of 'this one has a very masculine scent to it, it's the kind of scent I like on men.' You could see a sense of mini-panic in the main presenters eyes, and she quickly added 'Yes, although it would equally suit any women out there who like the smell of <whatever the stuff was>.' Then, a bit later in the show, they were selling a multi-pack of lotions promoted as being 'for men' and this featured on the tv screen blurb. Again, no sooner were they into the promotion when the main presenter said 'I know it says for men on the screen, but again I'd say anyone could use these ...' Whether they do the same thing when selling pills and potions aimed at women (i.e. 'men can also use these') I don't know.
My point is, it almost came across as though it was wrong to suggest something was more suitable for one gender than another. Yes I get all the stuff about inclusion etc, however I sense we're slightly moving into territory where referring to pure unadulterated full on masculinity or femininity is causing unease, not in all but some quarters. e.g. don't buy your son a blue coloured my first mini tool set and don't buy your daughter a pink coloured my first make-up set as you're imposing gender stereotypes on them. Everything's to be vanilla according to some people. That side of progress disappoints me. Maybe not a very good analogy, I have a young niece. She is very talented, bright, loves school and learning. I don't doubt she'll go far. She also loves the colour pink and is loving starting to muck about with make-up. To some (and no I'm not saying all) this is wrong. My sister is allowing the possibility of gender stereotyping.
And please, as some like to do on here, don't ask me to site references to back up my assertions. This is a general discussion forum on a diy website, not a university assignment.
As to the OPs question, I'd say 'no.' If we're aiming for true equality where everyone can feel comfortable, perhaps we should have male, female, male and female, transgender and gender fluid changing rooms and toilets, no?