They can't do that because a CMS under Cloudflare needs to be configured to bust the Cloudflare cache when content is edited or the user will see cached content after they edit a page.
OTOH a CMS that doesn’t send correct cache headers is already broken
That might be the default configuration if your CMS doesn't return any cache-control headers. But Cloudflare definitely [0] supports must-revalidate and etag or last-modified, which is also probably supported by whatever CMS you're using. HTTP conditional requests are very old and very widely supported.
[0] Okay, my claim is only definite up to my memory of using Cloudflare for a fairly high-traffic circa 2019-2022. I haven't used Cloudflare after that point, but a quick search of their docs shows support