> Correct me if I'm wrong, but doesn't Akamai have a not-insignificant minimum monthly charge?

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.

Need to create page rules if you want to cache HTML pages.

Also, cloudflare does support clearing of individual files. There is an API for it. Pretty sure their web client does too.