Why not simply reword the question to ask whether you prefer to be referred to as a "he" or "she"? In the rare case that it doesn't align with your genetic or anatomic gender, it's not exactly the end of the world.
Wonder what others think
However, I think doing things this way is going to be quite difficult to implement when you consider multiple languages. Male/Female/Other with an option to hide this information is much easier a solution.
I Don't think I'd ever use a "They" in a way such as being described here.
"John was late for dinner. They were caught in traffic."
Or is it "They was caught in traffic." ?
"John was late for dinner. They were caught in traffic." is perfectly correct usage (however if you'd just called somebody by a gender-specific name there's not much point in then being gender-neutral).
It sounds ungrammatical because the gender-neutral pronoun follows a gender specific name. The actual use is more like
"A friend of mine was late for dinner. They were caught in traffic."
The issue here is whether there should be an option to make the data non-public.
1. What gender do I want to identify myself as?
2. Do I want to publicly publish this information?
The issue is being muddied because some people that answer 'no' to question #2 are using the gender of 'Other' as a hack around the system, when in reality their gender identity is not 'Other.' This in effect co-opts the gender identity of 'Other' from the people that truly want to be gender-identified as 'Other.' Now when you view a profile with a gender of 'Other' you can't be sure if the person really identifies as 'Other' or if they identify as male/female, but are trying to hide it.
We could easily frame this problem in a different light without getting into the "OMG! There are really only two genders! I hate all this PC bullshit!" discussion. If Google+ made mandatory both choosing a gender (only a binary male/female choice) and publishing that choice publicly, then a lot of women that wish to hide their gender would choose 'Male.' Now the gender field is useless because you've designed a system that doesn't fit your users' needs, forcing them to provide you with wrong information.