Long ago "he" was the acceptable singular pronoun for all of humanity. Then culture moved faster than language. No one wanted to say mankind anymore. Man and womankind was too awkward. The word "humanity" didn't seem as objectionable since "human" seems to originate in "humus" meaning earth. It became too awkward to say "Everyone should bring his or her spousal unit to the ball." "They" and "their" became popular as a generic singular pronouns to be used when people of any gender are involved. They is gradually becoming accepted in certain cases in formal writing. Where it is still questionable is in the examples in my blog. If the subject of your sentence has been clearly identified as being an individual with a gender, by birth or by choice, it sounds awkward to say, "The man got their inflatable doll." If I understand what Meriadoc is saying, he or she seems to want to end all uses of genderized pronouns. I'm not quite there yet.
I see. I guess I still use "he/she" instead, because I feel like I ought to use a singular pronoun, but that does obviously indicate an obsolete binary. I understand that it's "becoming accepted in certain cases of formal writing," I suppose I'm just resistant because it seems like technically it should be incorrect. I see where the problem is though. We lack a truly appropriate pronoun.