I remember Cloudflare's job was to cache the page back in the day.. is it still the case?

It depends on how the site is configured in Cloudflare. I'm pretty sure I had to change the default settings to make it keep my site up even if my server goes down (and even then I'm not sure I did it right)

The default settings don't cache HTML. (Maybe even if the server says to cache it. I seem to need to add special rules to make it respect the standard cache headers)