In The Horse and His Boy, C.S. Lewis unwittingly provides evidence for a thesis that I think is likely to be true but isn't the sort of thing that you'll find much evidence for, one way or the other. The claim I have in mind is that singular "they" or "their" is not precisely gender-neutral. At least I'd say that the conditions under which it's most commonly used (and thus feels most natural) are sensitive to one kind of gender concern, and there are some instances where it would seem awkward to use it when you know you're speaking of one person whom you and the audience hearing you know the person is male. I think this is true to the point where I almost always assume that someone using it of an individual person to disguise the person's gender is almost certainly talking about a woman or girl.
I think it's well-established now that singular "they" has a long history in the English language and is not the result of feminist machinations in the last few decades, as I've heard some claim. It occurs in the King James Bible and in Shakespeare, so there's no arguing that it's a recent innovation. I also discovered it in C.S. Lewis, but what was interesting to me is that he doesn't use it consistently. He has two very similar sentences in two scenes near the end of the book, one from Aslan to Shasta and the other from Aslan to Aravis. When walking alongside Aslan to Narnia, Shasta asks him why he wounded Aravis. Aslan's response:
"Child," said the Voice, "I am telling you your story, not hers. I tell no one any story but his own."
Then later, when Aslan appears to Aravis, Bree, and Hwin, Aravis asks about what will happen to her servant because of her running away. His response:
"Child," said the Lion, "I am telling you your story, not hers. No one is told any story but their own."
Lewis, probably instinctively, refrains from using a generic male pronoun when generalizing from the initial case of a girl, when he did use the generic male pronoun when generalizing from the initial case of a boy. This isn't direct evidence exactly of my claim above, but it does suggest an unconscious gender-related difference in treatment in Lewis' writing, and remember that this book was published in 1952. He seems to have thought a singular "their" was inappropriate when the primary person being referred to was male but not when the primary person being referred to was female.