I really don't see the point of trying to add it to your account.
Why not just use it as a payment method when you check out? Avoid all these hassles and also then you could use it elsewhere.
The main problem with just using it as a payment method is that you can only use it on orders where the amount due is less than the balance of the card. Once you've spent it down to a smaller amount, it may be hard to find a small enough purchase. That's when converting the remainder to an Amazon card is really useful. Otherwise, Visa just gets to keep those last few dollars.
Sometimes you can use up a visa gift card at a store by splitting a payment onto two cards. You have to find a cashier who knows how to do it, and the store has to allow it (they pay an extra merchant fee for it), and it annoys the people in line behind you.