As a data point to
Mattg's post above, Ms Galah was denied purchase of an AC 30-day membership at the AA lounge in Chicago on the weekend, and was told "we don't sell those anymore". After pointing out that you could still purchase a 30-day membership online through aa.com, the lounge dragons were unmoved. So, Ms Galah had to sit there and purchase it online before showing the confirmation code to the dragon (still unimpressed at this point), who granted access.
Also note that it appears you need a desktop/laptop computer to fill out the membership form as the "start date" selector doesn't appear to work on mobile (iPhone at least)..