It's Amex's problem contractually. The contractual relationships here are:
customer -> card issuer (amex) -> card network (amex) <- card processor (stripe) <- merchant -> customer
So as you can see, you have no relationship with Stripe. Your have a relationship with Amex and the merchant. Those are your points of contact.
When you make a chargeback, Amex accepts this chargeback, and sends it to the merchant's processor, who then presents it to the merchant for response. This is a process defined in the contracts between each of the parties.
To your other question regarding fraud controls in this system, that's handled in the contracts:
- If you skip out on your debt, Amex must still pay for any charges they authorized. (The payment flow is customer -> Amex -> stripe -> merchant. This flow is reversed for a chargeback/refund.)
- If a merchant skips out, stripe is still responsible for any chargebacks.
Think about what this means: If stripe accepts too many high-risk merchants, they'll lose money. If Amex accepts too many high risk customers, they'll lose money. So they each have an interest in controlling fraud.
So what happens when a merchant gets too many chargebacks (typically less than 1%): Stripe will refuse to do business with that merchant. Why would they do this? Because if Stripe has too many chargebacks, the card network will refuse to do business with Stripe. They may be able to recertify as a high risk processor, but that comes with additional requirements... and if its above those high-risk levels, the card network won't allow stripe to process any payments at all.
This is all defined contractually.
What is the contract you have? You have a contract with Amex: your credit card terms. And you have either an implied or explicit contract with the merchant that they must meet.
Stripe and Amex are not fully aware of your contract with the merchant (refer to first relationship graph above). Part of the chargeback process is the merchant's response. A valid chargeback defense is that the charge meets the contractual terms the customer agreed to (assuming nothing illegal is going on). When the merchant presents the contract in their response, Stripe and Amex can review that contract. Amex (as the card issuer) gets to decide if they accept or reject the merchant's response and issue a decision on the chargeback. (Stripe (on behalf of the merchant) can disagree with this, and it then goes to the card network for a decision.)
So that's the whole process.
If you go to stripe directly (as a customer), you're attempting to do an end run around this contractually enforced process... and stripe isnt going to do that (unless they want to be sued for tortious interference by the merchant).
So hopefully you can see why Amex telling you to call Stripe is so bizarre. What makes it even more odd is that the type of dispute you have is something Amex handles like a 1000 times a day... they have a process for it.
And FYI: Amex can block a merchant from charging you in the future. (Easy on their part: just stop authorizing the charge from the merchant.)