"they style themselves 'singing circles' and make it clear that there is no restriction on what is sung
They seldom, if ever book guests but rely solely on local talent"
That, apart from the name, and that instrumentalists are also welcome, would be a fair description of my local "folk club." The fact that my local club has the self-contradictory name of "Broadside Folk Club" merely serves to point up the fact that, on this side of St George's Channel, "folk" is a term used fairly loosely.
It seems to me that you are only quibbling over the name, not the phenomenon, of English folk clubs.