Sherlock was not a club member, that was Mycroft (I vaguely remember him being one of the founders, too).
The rest is true though. With the addition of Sherlock having major bouts of depression where he would not leave him rooms for days.
It’s true that Doyle never mentioned it, but it’s logical that Holmes would want a place he could duck into for a quick meal; or a nap; or to put on a new disguise. Watson does say that Holmes had bolt holes around London, so it would have been foolish for him not to have a membership.
I remember reading about his membership in a non canon work.
If you’re a fan look up ‘The 7% Solution.’ It’s available as a novel and a movie; Holmes meets the only doctor capable of curing his cocaine addiction, Sigmund Freud.
