Right, but if you're doing a SPA, your SPA makes the login call and stores a copy of the session token in local storage, which unlike a cookie isn't automatical

Right, but if you're doing a SPA, your SPA makes the login call and stores a copy of the session token in local storage, which unlike a cookie isn't automatically sent on any request, never mind cross-origin ones. Doesn't prevent against XSS of course, but then that's what CSP is for.