Singular "they" sometimes gets a bad rap (less so in the last couple years) but it's been part of the English language since the time thorn (Þ) was an English letter. It's appeared in well known written works by respected authors, such as this usage by Shakespeare:
"There's not a man I meet but doth salute me as if I were their well-acquainted friend" - A Comedy of Errors, Act IV, Scene 3
As you point out, even people who criticize it use it, because it's natural in most situations. There would be very few people who would naturally say "each of the children called his parents" over "each of the children called their parents."
https://www.npr.org/2016/01/13/462906419/everyone-uses-singu...
>A lot of people tell you it sets their teeth on edge, but my guess is that they're not listening to themselves very attentively. Everyone uses singular "they," whether they realize it or not. In an engaging recent book called Between You & Me, the New Yorker's self-designated comma queen Mary Norris says that that use of "they" is "just wrong." But flip back a few pages and you find her writing "Nobody wanted to think they were not essential." If that sentence got by a New Yorker copy editor in her own book on grammar, what hope is there for anybody else? If you insist on coming down on singular "they," it's best to be a bit ironic about it, since you're almost certainly going to be making a hypocrite of yourself before the day is out.