For many years I used ze/zir for everyone, and fairly recently I changed to they/them (still working on it, I have all of the slip-ups). I made the switch for several reasons. One is that even though ze/zir does not refer to a gender and is thus a term for any person, it is taken as meaning "something other than he or she" and I have to acknowledge the common perception even though it's inaccurate. So I use they/them because that is actually taken as gender-neutral. There is also the fact that I am not the clearest enunciator, and people sometimes hear my "zir" as her and my "ze" as he. That doesn't work either! And lastly and most importantly, they/them is inclusive of people who are bigender or multigender and plurals (people who are multiple systems, multiple people living in the same body. FAQ). And because of the misconception about ze/zir, using that set for all people can be hurtful to binary trans people because it may be perceived as stripping gender, but "they" is less likely to be perceived as an invalidation of identity. So, overall I switched to 'they' to communicate more accurately and to be more inclusive.
Also, I realized recently that as a person who only feels mild discomfort at people using gendered pronouns about me, I am relatively privileged, and I need to mobilize this privilege for the sake of those who feel deeply wounded by gendered pronouns. So I have decided to require people to refer to me in gender-neutral terms, because that will make it easier for others to require it when they need it. I am not entirely sure how to go about this, because I haven't done it before, but it's my intention. Please poke me if you notice me failing to do it; that will help me to remember and/or have the courage to do it. And if you do not care what pronouns people use in reference to you, I urge you to also require gender-neutral from those who refer to you. You won't be hurt if people tell you no, so your risk is much smaller and you can make the world a safer place, in a small way. You never know who around you is wincing inside every time someone says "he" or "she."