Henry Fielding: "Every Body fell a laughing, as how could they help it."

Thackeray: "A person can't help their birth." (spoken by Becky Sharp, the main character, with reference to her own social climbing)

George Eliot: "I shouldn't like to punish anyone, even if they'd done me wrong."

Lewis Carroll: "'Whoever lives there,' thought Alice, 'it'll never do to come upon them this size: why, I should frighten them out of their wits!'"

I don't see how anyone could read such examples of singular "they" and not immediately be relieved of any objection they may have been entertaining. The extraordinary thing is how modern it sounds, even when the rest of the language is archaic:

"Every one Sacrifices a Cow or more, according to their different Degrees of Wealth or Devotion" (Dr. Johnson)

"Let nothing be done through strife or vainglory; but in lowliness of mind let each esteem other better than themselves." (King James Bible)

Sometimes the formulation seems deliberately mischievous:

C. S. Lewis: "She kept her head and kicked her shoes off, as everybody ought to do who falls into deep water in their clothes."

Doris Lessing: "And how easy the way a man or woman would come in here, glance around, find smiles and pleasant looks waiting for them, then wave and sit down by themselves."

E.B. White: "But somebody taught you, didn’t they?”

That last one (from Charlotte's Web) is delicious because White put the White in Strunk and White and knew full well when a writer should break his own rules.