No, Sandy. Members and staff of the Athenaeum are too bound by convention to tell anyone anything, but they wouldn't be protected. There's a distinction, which I'm sure the judge would acknowledge, between a club which may have a convention (not one I'm aware of, by the way) and an AA meeting where people are invited to confess to misdeeds on the explicit understanding and rule that their speech would not be reported outside. There is no absolute and binding rule or explicit understanding that what one member of a gentleman's club says in the club is not to be repeated elsewhere.
Yes, it's a question of fact and degree. You'd have a job keeping out a statement in a confessional that the congregant was going to leave the box and kill his wife, in the subsequent trial. Similarly, something found in a house search conducted without a warrant, if one was technically needed, but which had not been obtained in time, might well be allowed in. But judges are quick to exclude admissions made in consequence of breaches of the rules on police interviews, since they have a deep suspicion of the merits of such evidence anway and think that the police should not be allowed to break rules devised to protect the public against such abuses. The law is against such practices.