You are probably technically correct but this one has been going since 1912 and has been viewed as THE supporters club for a very long time.
http://www.bhasc.com
I’ve said for a long time that the official supporters club should do more but it’s a total waste of time. Full of old cronies and they make no effort at all to get new members. Last time I asked it only had 300 members.
So yes if a vibrant new independent supporters club was formed I would join.