It seems to me that in this case 30 seconds of Googling the org by a properly trained human moderator would have prevented the takedown and saved them yet more embarrassment concerning their rinky-dink excuse for a content moderation system.
Yes, multiplied by millions of legitimate and bad-faith requests that means a lot of man hours and a lot of money, but that’s one of the prices you pay if you don’t want your international media brand to be considered a cesspool of disinformation and bigotry.
That would be an interesting approach to cut the Gordian Knot but I doubt that FB would be willing to accept the regulation (including anti-trust provisions) that goes along with common-carrier status. As you say, they want to have things both ways.