If it’s purely static, then a simple “Cache Everything” rule should get you almost everything you want. It’ll be cached globally, and Cloudflare already handles DNS and things globally.
If you wanted some extra oomph, Argo could aid in certain distant regions for the initial hit to your server, but after it’s cached, it’ll have very little extra impact.
If it’s purely a static site though, you should consider just hosting it on Pages. Then you won’t have to worry about anything and Cloudflare will serve your site globally, with the all the best pre-configured settings, for free.
It’s possible your site has low traffic and items are being evicted from the cache. Or from varying regions. You may also need to respond with a cache-control header, or configure the edge/browser cache TTLs. If you can share an example URL that isn’t cached, I’d be happy to take a look.