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The real reason is that they don't want the paparazzi hanging around and they really don't want the scrutiny into the club's details (like membership lists, past member affiliations, etc) that Ms. Swift would bring along. It's not really a new vs old money thing, but that does make for a better story.

Pretty much its the case EXCEPT maybe the Hamptons, and a few very old (read, direct descendants 'spiritually' to their English colonial forebears) institutions, but the list of spots like that are quite rarified air compared to the UK.

I do believe this is the bes explanation.

This also happened when Michael Jordan tried to join country club/golf courses in the wealthy northern suburbs of Chicago