Cookies don't show up in developer console of Safari/Chrome with Set-Cookie Header, but exist when looking at server logs
Have you tried opening a tab to the server https://*.amazonaws.com and checking there instead?
The cookie will be set on the server's domain, but you won't see it in your local server's cookie storage. The reason is that all web storages are bound by same origin policy and your document can only access storages from its own domain, and the server can only set cookies for it's domain.
The rationale here is that if I sent you a link to a rogue document, it can't exfiltrate your SO cookies even if they were accessible from JS, neither sending a request to a rogue server can overwrite cookies on SO.
Try to disable chrome://flags/#network-service and it should work properly.