Yes I believe public bathrooms should be open to people of either gender. Because we need to ask why they are separate in the first place. There are privacy stalls, so I don't see the issue, unless one is exposing thier genitals to others, in which case biological gender would be far more important than how one identifies.
If one goes to a doctor, biological make up is more important than how one identifies.
If I'm hanging out with someone, thier gender is (or at least should be) irrelevant. If I'm hanging out with biological females, we might be more likely to commiserate over periods and cramps (Biological issues that have nothing to do with how one identifies).
I was mainly adressing the idea that there is no scientific/biological basis for more than 2 (or three, if you count "other") genders.
I think inventing new labels (particularly ones with no basis in science) to make it easier to categorize people who do not accept traditional social roles (social roles have no basis in science, it is something we have merely accepted as a culture, like religion) is more sexist than not using them at all. The reason we keep coming up with "new genders" is when someone doesn't fit neatly in the social boxes we've already invented, we invent a new box. I say we get rid of the boxes altogether.