AAA's main reason for not wanting people to charge things on credit cards is that if they bill someone once a month, they make more than 10% more money off of them: their finance charges are horrible. (Since you're paying off the money gradually over the course of the year, and you're being charged 10% of the total amount instead of the outstanding amount, it's a lot more than a 10% interest charge would be from a credit card.)

Whereas if someone puts the full amount on their credit card, someone else is getting that money. The last thing AAA wants to do is make it convenient for people to pay off their entire bill.

There's a large lawsuit over these financing charges going on, I seem to recall.