The great thing about static sites is that it's really easy to put them behind a CDN. Amazon Cloudfront is pay-what-you-use and seems pretty reasonable especially for smallish sites.
And does Cloudflare cache HTML pages? Last I checked it only cached (what it considered) static assets and hit the origin for every page.
Yes. "It was like that when I got here." I'm working on it.
> And does Cloudflare cache HTML pages? Last I checked it only cached (what it considered) static assets and hit the origin for every page.
Yes. This is configurable.
Also, cloudflare does support clearing of individual files. There is an API for it. Pretty sure their web client does too.