I assume transmissions by non-controllers is prohibited by internal FAA directive.
Not so clear. The
FCC controls radio communication. The FAA controls aircraft and airspace. I doubt that there is a FAR part or FCC part that specifically prohibits this behavior.
If there is administrative action against the controller it will be under one of the catch-all, subjective, regulations covering "interfering with the security of air operations". I predict it would be upheld by an administrative law judge and overturned by the entire board. If it gets that far.
There is no harm, no foul in this case. Management would be covering their @ss while waning Controllers to not get caught by FAUX news in the future.