I have had the card a couple of months and have entered clubs 13 times (DFW, BOS, PHL, DCA) and have never been asked for a boarding pass. I have been asked for ID once.
On this last trip, I did try to enter with my BP (which was AA) just to see if my access was linked to my AAdvantage number, but that did not work.
So my experience is showing the card is enough. The bigger issue is having the agents figure out where the dumb mag stripe is on the card.