Earlier quoted context omitted.
> ending ties with anti-gay groups. Salvation Army. The Fellowship of Christian Athletes. Calling those groups "anti-gay" and protesting a fast food restaurant for donating to them is absurd.
Could you please elaborate on why those groups aren't "anti-gay"? I've never heard of "Fellowship of Christian Athletes", but the Salvation Army is absolute anti-gay.
To illustrate, I might refer to an organization under religion X as an "anti-women religious group" because perhaps religion X had some policy on women's activites that I believed were sexist; then for me "anti-women religious group" would be a technically correct phrase in that X is "anti-women" and "a religious group", but the phrasing would probably be objectionable to practitioners of the religion who don't see those particular policies as fundamental to their religion, and perhaps didn't even consider the policies sexist in the first place.
In general, I don't think I often see this loose use of terminology applied to other cases; like I don't think people call Nike a pro-child-labor company, although if that were the topic of discussion I might say Nike uses child labor. To me it's about avoiding ambiguity.