Lee,
Your questions always make me think.
Con: It would be a nightmare of paperwork, and would probably close said club down shortly. The only pro I see is that it would definitely cut down on the "looky loos".
On the other hand, having a testing contact available for members would be a perk. Some folks are shy about asking their doctor for one. Just give them the number for your contact or have their business card somewhere, and you step back out of it. They can handle it in private.
As for "private groups", that's another story. If you had a private group who agreed, then it might work.
Mrs. D