I'm not sure that's as true as it used to be. A lot of great modern British comedians weren't members (though may have been Oxbridge): Chris Morris, Stewart Lee, Charlie Brooker, Armando Iannucci, Bill Bailey, Simon Pegg, Steve Coogan, Sacha Baron Cohen, Ricky Gervais. There is an argument that Footlights continues to dominate a particular strain of "awkward Englishness" style comedy, i.e. Fry & Webb, Hugh Laurie, and to a lesser extent Richard Ayoade. But I don't see these as the defining characters of British comedy at the moment.