The prescription:
'they/their' is plural so it can't be used with a singular antecedent such as 'everyone to refer to people in general.
The reality:
As with all prescriptions, they contain no sense of reality.
Let's start with an example:
Everyone brought their sleeping bags.
This, according to the prescriptivists, is illogical because, they say, 'everyone' is singular but 'their' is plural.
This notion is illogical on a number of grounds.
First: collocations like this, 'everyone - their', have been in use for hundreds of years.
Second: everyone intuitively knows that 'everyone' is grammatically singular but notionally plural. There isn't a native speaker alive who would think that upon saying to an audience of a thousand or a million people,
"Everyone, please stand up",
that one person would stand, leaving the speaker to repeat it a thousand or a million times.
Third: both the words used in English and the grammar structures do double, triple or multiple duty in the language. There are so many of these, it's a wonder how the prescriptivists missed them.
{I'll let you in on a little secret - very little to no thought has ever gone into these prescriptions. They have arisen for the most spurious of reasons and perpetuated themselves for the same reason, very little or no thought given to their veracity]
As I say, there are many but let's just deal with the one that completely blows this prescription out of the water. It's a FACT that 'you' is used both as a singular and a plural. In fact, the pronoun 'you' in the singular is used with plural verbs, 'are' and 'were', just as the plural 'you' is used with the plural verbs 'are' and 'were'.
There are many more but that will suffice for now.
By the by, never ever use the Chicago Manual of Style as a reference for grammar. That would be like using Strunk and White, or Harbrace College Whatever, the Capital Community College/Darling website or Brians Errors or the Grammar Girl.
They all get some things right but they are so rife with errors that you're much better off seeking a reliable source, eg.
Merriam–Webster's Dictionary of English Usage
===========================
A few comments on generic 'he'.
1) Everyone brought their sleeping bags.
2) Everyone brought his sleeping bag.
Is 'he' really generic, does it work? Look at sentence 2). Prescriptivists are big on telling us that prescriptions are intended to make language clear and concise.
So in #2, doe it appear that everyone/all the people carried one sleeping bag,
his sleeping bag? How silly is that? Did everyone/all the people also sleep in the same bag.
Okay, let's fix it. We'll just make it a plural.
3) Everyone brought his sleeping bags.
Is this sensible? Sure, it is, if you want to say that 'he' supplied sleeping bags for everyone/all the people.
Now look at sentence 1),
Everyone brought their sleeping bags.
Much more sensible - no mental or grammatical gymnastics are needed, there's definitely no need to recast the sentence - as I said, native speakers haven't seen any need to recast this perfectly natural collocation since, well read it for yourself.
Quote:... it
[The singular "they"/"their"/"them"/"themselves" construction] actually dates back to at least the 14th century, and was used by the following authors (among others) in addition to Jane Austen: Geoffrey Chaucer, Edmund Spenser, William Shakespeare, the King James Bible, The Spectator, Jonathan Swift, Daniel Defoe, Frances Sheridan, Oliver Goldsmith, Henry Fielding, Maria Edgeworth, Percy Shelley, Lord Byron, William Makepeace Thackeray, Sir Walter Scott, George Eliot [Mary Anne Evans], Charles Dickens, Mrs. Gaskell, Anthony Trollope, John Ruskin, Robert Louis Stevenson, Walt Whitman, George Bernard Shaw, Lewis Carroll, Oscar Wilde, Rudyard Kipling, H. G. Wells, F. Scott Fitzgerald, Edith Wharton, W. H. Auden, Lord Dunsany, George Orwell, and C. S. Lewis.
http://www.crossmyt.com/hc/linghebr/austheir.html#X1a
If there had been a rule from the beginning of language that prohibited 'they/their/etc' from being used with a grammatically singular element, then maybe it would have a measure of veracity. It doesn't. This "rule", like all the prescriptions were invented/concocted/made up.
Quote:Singular "their" etc., was an accepted part of the English language before the 18th-century grammarians started making arbitrary judgements as to what is "good English" and "bad English", based on a kind of pseudo-"logic" deduced from the Latin language, that has nothing whatever to do with English.
Note the underlined part at the end, "nothing whatever to do with English". That's the underlying rational for most every prescription. That's why people don't follow these nonsense rules.