The login information isn't so much crossing boundaries -- evil.com can't read your session cookie on bank.com -- but cookies that don't set a SameSite attribute allow anyone to send that information on your behalf, and effectively act as you in that request. Textbook example of a "confused deputy" attack.
> What stops a script on evil.com from going to bank.com to get a CSRF token and then including that in their evil request?
The token is either stored on the page that bank.com sends (for a html form) or was sent in a header and stored in local storage (for API clients). In neither case can evil.com read that information, due to cross-origin restrictions, and it changes with every form.