I was typing a response wondering why gay people get mad over the word fag, but I think I thought of why.
It is the same reason why black people get upset over the word "Nigger". Nigger was used because "niger" means black in latin. The word "nigger" doesn't become offensive until it is used by the people denying african-americans their rights in the 20th century.
Maybe gay people do not like being referred to as "fag" because they are being denied the right to marry or adopt. So it could seem to them that "fag" is a label for persecution.
Or I could be way off cuzimnotgai.