The Admirals Club is a membership lounge, not a business or first class lounge. As such, while you may be allowed access to the club based on the ticket you are flying, wifi is one thing they reserve for members. The members pay hundreds of dollars a year for access to the club. Also regarding access via the AMEX cards, The American Express agreement pays for access to the club, but not wifi. You might contact American Express and ask they why but of course you already knew that.