Actually, I think there was quite a bit of what you might call Country in SOME of the folk clubs in the 60s, stretching into the 70s.
I've just had a look through the guest list for the Surbiton Club (which was particularly eclectic at the time) - I'm surprised John Mackenzie doesn't remember as he was around quite a bit in those days. It's true that a lot was at the Bluegrass end of the spectrum - Southern Ramblers, Orange Blossom Sound, Pete Stanley, Malcolm Price etc. - but I also found some names that were more mainstream Country - Johnny Duncan & Pete Sayers, George Hamilton 1V, Bill Clifton, Doc Watson & Ralph Rinzler, Mike Seeger, Hedy West, and quite a lot of the visiting Americans of the day had influences more from Country than what you might call American folk. People like the Carter Family influenced many folksingers of the day.
I was just this Summer talking to somebody about how much more eclectic things were back then. We, at Surbiton, had all of the above, plus a lot of Blues - JoAnn Kelly, Rev Gary Davis, Champion Jack Dupree, Jesse Fuller, Gerry Loughran, - and Jazz - Ken Colyer Jazzmen, Diz Disley, Stephane Grapelli. As well as all the singer/songwriters and strictly Trad - Stewarts of Blairgowrie, Isabel Sutherland, McPeakes - and all the revivalists doing trad material.
I venture to suggest that you wouldn't get this nowadays.
Many of the performers fell into multiple 'camps' and it is quite difficult to say where the boundaries between, folk, country and bluegrass actually lay, but I still think this was a remarkable scene back then. We all learnt so much about the rich sources of folk music.