Using a cookie to provide the CSRF token to the client does not allow a successful attack because the attacker cannot read the value of the cookie and therefore cannot put it where the server-side CSRF validation requires it to be.
The attacker will be able to cause a request to the server with both the auth token cookie and the CSRF cookie in the request headers. But the server is not looking for the CSRF token as a cookie in the request headers, it's looking in the payload of the request. And even if the attacker knows where to put the CSRF token in the payload, they would have to read its value to put it there. But the browser's cross-origin policy prevents reading any cookie value from the target website.
The same logic does not apply to the auth token cookie, because the server is expects it in the request headers and the attacker does not have to do anything special to put it there.