As an intrusive outsider, let me put ni my two cents.

"English" in this context might mean several different but perhaps overlapping things.

It might mean a Society interested in folksong and dance but made up of only English people.

Or a non-restrictive FS & D society based in England. (As I've always interpreted it.)

Or a society devoted solely to English FS & D.

(And for added interest, recall that at the time of its organization, "England" was in frequent use - in England and the U.S. at least - as a synechdochal synonym for "Britain.")

So perhaps the Society should keep its name, while attending to any aspects of FS & D its members prefer.

Just sayin'.