One issue is their not for profit status. That may seem a little ridiculous when you consider that the club probably isn't making any money and could be operated to ensure that they don't, but there could be problems with the ability to restrict access to certain groups, and maintain certain rights for membership etc.
I don't want to have a political discussion about these issues, but once pandora's box is open....it is very hard to close it again in better times. That could be a drawback for some clubs.
Of interest, I belong to a club where we created what is really a semi private group of members. They have provided a nice infusion of capital for the last 4 years. Paradoxically, the cash they infused may have delayed us making some very difficult decisions sooner than now under a very difficult economic environment.