It is theft, but it's not amazon stealing it, it's somebody else. Amazon knows there is somebody scamming them, but, here's the kicker, they don't know if it's the caller complaining or the person placing the order.
The best thing to do is exactly what they said, file a chargeback and eventually Amazon can take action.
You'll notice the person running the website never cooperated with answering Amazon's questions, it's no wonder they didn't work with her. If you don't want to work with the merchant, then contact the credit card company. Not hard to figure out.
At a minimum Amazon should have to help clearing up the issue. I guess some entities are above the rules most of us have to live by. Same with hospitals that can repeatedly charge bogus items without any consequences.
The trick is balancing account privacy with fraud prevention. Without being the account owner or having any real relationship with Amazon OP has a hard task of proving who they are and generally you wouldn't want Amazon sharing any account information with non account owners.
I think the real solution would be for them to just provide an auth mechanism for people to prove they are the CC owner so they can remove the payment method from any accounts and not give out any info on the name or email on the account. I've seen a few things that use a small charge that gets refunded and you enter it back in the app as a 2FA.
Just because they won't tell your they're looking into it (because they don't know you or you're not authorized to discuss someone else's account) doesn't mean they're not taking action behind the scenes.
It's not their job to secure your card outside of their own system. Assuming these cards aren't stolen from within Amazon's own servers, it's up to you and your credit card to protect it from unauthorized use.
Yet there's this alternate universe where there are laws against facilitating a crime, receiving stolen property, etc. I'm not saying that sort of thing applies to Amazon, just saying isn't it weird that it doesn't?
A traditional method used was to send two very small random payments, say 0.03 and 0.09 to the card in question, then ask the caller to confirm the payment details, how much etc, once they'd arrived. If they could provide these details that is pretty much as far as they can go in terms of verification.
And if the card has two authorized users, with the second person making purchases the first one doesn't know about? How is that at all Amazon's problem?