The AAA card is for those "It's so broke there ain't a damn thing I can do with it sitting here" situations. Like the night of my tranny puking the rear countershaft bearing - A clear case of "Goin' *NOWHERE* fast", and gonna stay at exactly that speed until either the car gets to where the spare tranny and "full shop facilities" are, or the tranny and shop comes strolling to the car - and you can damn sure bet THAT ain't gonna happen anytime soon!
For cases like that, you're going to be sitting there anyway (unless you choose to start hoofin' it someplace - which might be a pretty damn stupid stunt for any of several reasons) so it really doesn't much matter how long the wait is going to be. Might as well find someplace to dial "1-800-AAA-HELP" then make yourself as comfy as possible 'till the big yellow truck pulls up.
Of course, counting on AAA as your "first option" when something goes wrong - especially if it's something you can bailing-wire-and-bubble-gum into working long enough to get you home/to the shop - is just plain nuts, but for the situations where you've got no choice... Well, that's why I pony up my 80 bucks a year - I *KNOW* that no matter how broke the car is, it (and me) *WILL* get to someplace that can make it "unbroke" - or at least declare it "really, truly dead".