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