So, commenters, is it that all these writers (whose work ranges from the late 1500s to the 1900s) and many more were wrong, and you’re right, when you say that “their” can’t be used in these contexts? Is it that you have the Logic of the Language on your side — the same logic that tolerates the singular “you are,” “aren’t I?,” “ice cream,” and much more, but that as a matter of the laws of logic balks at a singular “they”? Or is it just that you’re discussing what you find aesthetically pleasing (or even pedagogically optimal, for instance with an eye towards teaching students usage that will satisfy self-described “purists” and will thus serve them well socially)? If it’s the latter, I’ll happily end the debate. But my sense is that many people who denounce the singular “they” (including where the singular relates to nouns with a collective meaning, such as “everyone”) and similar matters are making an assertion about correctness, and not just about their own tastes or about the most useful teaching approaches.
As a writer -- of blog posts, personal correspondence, scientific papers, and the like -- my style priorities come in this order.
- Elegance (a context-dependent subjective judgment)
- Logical construction
- Adherence to standard grammatical rules
Most of the time, these four goals align. Occasionally they don't. And when they do, a writer has to make a choice. Everybody must take his or her own bag adheres to rules, and is logical. It is, I think, just as clear as Everybody must take their own bag. In most contexts it's not as elegant.
As a teacher, though, I'm a big believer in instructing kids and other emergent writers "slavish adherence to standard grammatical rules," followed by "logical construction of sentences."
First, I want them to learn the rules; occasionally context demands slavish adherence to standard rules, and in most contexts nonstandard writing creates unintended distraction rather than intended style and elegance.
Second, it's easier to teach younger children objective, clear rules. It's much harder to explain concepts like "elegance" and "clarity," at least until they're older.
Third, subjective evaluation of writing is made easier over time as learners gain confidence and comfort with the language they're writing in, and as they gain exposure to many contexts and to examples of really excellent writing that may play looser with standard syntax.
Fourth, you get more leeway with nonstandard writing when you're good at it, and especially, good at judging when it works and when it doesn't.
That's pedagogy for you -- and not everyone with an opinion on the subject has bothered to distinguish pedagogical from stylistic issues.