[MayflowerMadam]

Except that here, it’s not so much the word, “themself / themselves” (although it’s marginally bad). It’s the “s” at the end. It makes it a plural when the subject is singular. There’s no agreement.

[Tired of Taxes]

The AP is the problem. At the link shared by GenXPolymath, themselves is now used as a singular pronoun.

Still, a good writer could write the story with no pronouns at all.