Ok, it seems like maybe this isn't something that Visa or MC do. For the people asking about AGCs, here is what I do:
1. Buy a $1000 (or other large denomination) AmEx GC.
2. Call the number on the back of the card (I think it's 877-297-4438) to register it with your name and address, because you can't register AmEx cards online.
3. Cash out the card on an outlet such as AP.
4. Buy smaller denomination AmEx GCs--Staples, for example, only has $200 cards where you can get your 5x Chase Ink bonus.
5. Call the phone number again and talk to a representative. Tell them you have some gift cards, and you want to transfer the balance to another gift card.
6. You can only transfer up to the amount that's on the original card. In this example, you are only able to transfer 5 $200 cards onto the original card.
The whole transfer process takes at most 5 minutes. I have never run into problems doing it, and there's never a wait to get to a representative.
As to why I do it, some people have already touched on the reasons.
1. I don't want to raise Amazon's (or anybody else's) suspicions about why I keep swapping out payment cards.
2. Also, it's kind of a pain in the ... to continuously add new payment methods and delete them later.
3. I generally have to register the cards for online purchase anyway, so this process saves some time.
If anybody has specific questions about this, please feel free to PM me.