Exactly! It was used in phrases like "To each their own." but never "I will go with Pat to their birthday party."

"But Chaucer!" is a half-truth.

An incorrect statement is not a fallacy. An incorrect statement can indicate a fallacy's presence, but in this case I don't believe it does.

The fallacy in this case is a strawman. They are saying that if singular they is acceptable in certain situations (e.g. this very sentence), it then becomes acceptable in any situation (e.g. "Their name is Charles.")

The reality is that singular-they in these new variations is not a linguistic phenomenon, it is instead a political shibboleth.