If you need to authorize every single request (even to a static file) which means that every single request is unique, then this is obviously not a good use case for an edge server cache. You can still cache things in the browser with cache headers and continue using the CDN to proxy the full request with cookies to the origin. This doesn't add much latency and can sometimes decrease it because the CDN will keep faster connections open to the origin.

However, most CDNs today also offer their own access controls either with cookies or url tokens so you can do authorization at the CDN edge instead of the origin.

And yes, you can use the CDN to cache static files and just proxy requests to pages. Usually private information that requires authentication is in the webpage and the static files like javascript, css and images dont need protection.