One of the woke anti-intellectual counterarguments to support "their" or "they/them" as an identifying pronoun is that it is "neutral."

"They" is not neutral in the sense of grammatical number.  "They" can't be singular.  It (referring to the word "they") is plural.

Strunk & White's The Elements of Style is the Holy Grail of grammar advice.  Touring a fancy private school a few years ago, my group observed a high school English Literature class.  The students had read a Charles Dickens work and were writing essays on their individual laptops.  I asked the teacher what grammar references the kids used.  He responded that they (the kids, plural) simply used apps on their computers.  The teacher looked surprised when I asked the question.

This was problematic to me, and I certainly wasn't going to pay thousands for my kid to not know a consistent grammar resource.  The internet changes — stealthily changes.  One day, "recession" means one thing; the next, it is something else entirely.  One must be incredibly careful with internet resources, so it is always handy to have the age-old trusty books with pages by one's side.

Regarding pronouns, I wouldn't be surprised if today's style guidelines from the internet have been edited to favor woke grammar.  And kids therefore think "their" is a neutral method to refer to someone who feels neither male nor female.  But here is the real rule, straight from William Strunk, Jr.'s and E.B. White's Elements of Style, 4th Edition:

Page 60:

They. He or She. Do not use they when the antecedent is a distributive expression such as each, each one, everybody, every one, many a man. Use the singular pronoun.

Every one of us knows they are fallible.

Every one of us knows he is fallible.

A similar fault is the use of the plural pronoun with the antecedent anybody, somebody, someone, the intention being either to avoid the awkward he or she or to avoid committing oneself to one or the other. Some bashful speakers even say, "A friend of mine told me that they..."

The use of he as a pronoun for nouns embracing both genders is a simple practical convention rooted in the beginnings of the English language. Currently, however, many writers find the use of the generic he or his to rename indefinite antecedents limiting or offensive. Substituting he or she in its place is the logical thing to do if it works. But it often doesn't work, if only because repetition makes it sound boring or silly.

Consider these strategies to avoid an awkward overuse of he or she or an unintended emphasis on the masculine.

Use the plural rather than the singular.

The writer must address his readers' concerns.

Writers must address their readers' concerns.

For other strategies, order the book.

Pronouns (and grammar in general) are meant to provide clarity.  "Their" refers to a plural antecedent, not a singular one.  According to the be-all-end-all resource of grammar and style, this is clearly the case.  So yes, a singular person insisting on being a "their" or "they/them" is grammatically incorrect.

Strunk & White hasn't failed me yet.  I just wish the schools still used it!

The author would like to thank her high school English Teacher, Mrs. Baker, for providing her with one of the best resources for life ever written: The Elements of Style by Strunk & White.

Image via Pxhere.