Harvard grammar gripe

This snippet comes from a document entitled "Supplemental Style Sheet for Social Sciences" on the Harvard University web site:

Or, better yet, use what native English speakers have been using for hundreds of years: "they" or "their" as the singular generic pronoun. For example, "Everyone should put on their coats."

The sentence sounds right and the meaning is clear. Prescriptive grammarians tend to get upset at the use of "they" and “their” as singular pronouns, but they conveniently manage to overlook the fact that "you" and "your" are used as both singular and plural pronouns all the time.

Arg. "Everyone should put on their coats" certainly does not sound right to this native English speaker.

"Everyone" is singular, "their" is plural, and ne'er the twain shall agree. The fact that "you" can be singular or plural is not relevant.

Indeed, "Everyone should put on his coat" is unnecessarily masculine, and "Everyone should put on his or her coat" sounds awkward; but I prefer either of these to the ungrammatical "Everyone should put on their coats" which Harvard's style sheet endorses. Better yet, avoid the issue altogether: "People should all put on their coats."

(Note that the "Supplemental Style Sheet for Social Sciences" is on the Harvard web site of lecturer Don Ostrowski. I do not know if it applies to Harvard social science students generally or Ostrowski's students specifically.)