My experience has been different.
When booking from wife and daughter's AA accounts I've had to use their credit cards on line and over the phone.
Computer and agent both said no to mine.
Has this changed recently?
They will tell you that because that's genuinely what they believe. They were probably told that through training, etc.
But they have no way of verifying whether the name/address given matches the address or phone number or name on the credit card. As long as number, exp date and security code are valid and your issuer is not blocking the transaction and their is available credit then the transaction in my experience will always go through.
I have told them many times when paying that my card has the exact same name as my advantage account when it fact it is completely different. The transaction has gone through and the passengers got tickets issued and have already flown.
Once I got in a bit of an argument with an AA agent who insisted and questioned that the address that I gave her must be the same address that I gave my credit card issuer when I applied for the card. This was when I was new to AA redeeming. So I tried the best tactic I know, thanked her politely then HUACA (hung up and called again). The next agent was all good and flights were booked without a problem.