>"They" is also often used as a neutral form but it is incorrect (ungrammatical) when used to refer to a singular entity.
Simply wrong. There is no reason to support this assertion; it's a modern invention with no reasoning behind it - simply an arbitrary decision by a handful of grammatical prescriptivists who choose to ignore the large corpus of historical English text, not to mention the overwhelming current usage.
Since we mostly hear it coming from Americans, I conjecture that it may originate in Strunk and White (a highly questionable but ubiquitous American grammar guide).