Has anyone ever declared themselves to be male in the first person, female in the third, and neutral/plural in the possessive: "he/her/their"?
1) Because that's how people do it. Sometimes we all collectively settle on a standard that isn't the best, but that's just how it's done.
2) It allows for more easily expanding to multiple pronouns, e.g. "he/they" or "she/they", which indicate that the person is ok with "she/her/hers" or "they/them/theirs".
3) It makes it easier to state neopronouns (e.g. xe), by standardizing the form -- he/him/his is the same format as sie/hir/hirs or xe/xem/xems.
4) (My opinion) It feels more natural to say "he him" than just "he". "What are your pronouns?" "He." I dunno, it's a short enough utterance that it could be easily to mishear?
You will sometimes see people write something like "he/they", but that just means that either are acceptable.
If you're saying you personally don't care what gender you get referred to as, that's your right. But for other people getting misgendered can be a significant problem. If you want to change that, it's not those people you need to get to change. It's the ones making such a big deal about other people's gender.
If they're out of the norm then it may make sense to mention it then, I don't think society is hostile to it. I believe it applies to less than one percent of the US population, so it seems excessive to establish a norm of behavior for all, since harmony seems achievable despite that.
Of course the arrangement of my internal organs is relevant to doctors and medical sciences. As is my blood type and my cholesterol level. That doesn't mean you should be asking for my blood type when I'm registering for an online service (unless, of course, that service involves blood donation or something). That's the whole point -- it's really, genuinely, just not relevant to the vast majority of cases where it's asked.
On the one hand, you can't reasonably force people to select either masculine or feminine, particularly in English. On the other hand, if someone has answered the gender question with "neither / other" or "unknown / I prefer not to say", how would you refer to them in French, say? Would you ask "êtes-vous heureux", or "êtes-vous heureuse"? Unfortunately many languages do not currently have an established or convenient way of referring to non-binary people. (Though perhaps it's true that "most" languages are like Finnish or Hungarian in that gender does not feature in the grammar at all. The Indo-European and Semitic languages with which English speakers are most familiar are somewhat exceptional in that respect.)