When I surveyed static hosting providers about a year ago, the only ones I found that didn't insist on taking over your CI pipeline were Netlify[0] and Firebase. I assume it's possible on Cloudflare and AWS, too, but I wanted to go with a provider where static hosting is core to their business.
I've been happy with Netlify, but given how deeply they bury their "manual deploy" feature, I fear the day that they decide to force everyone into the Netlify-native build workflow.
[0] https://docs.netlify.com/cli/get-started/#manual-deploys
It largely makes up for their really basic build environment. Last I tried it you can basically pick between Ubuntu versions with some pre-installed software. If you wanted to use an unavailable tool you could either manually download it or you were out of luck.
Genuine question: and what would you pay (monthly/yearly) for such service? Let's say CDN serves all visitors from EU/NA.
I developed a service for similar purposes last year (I became tired of S3..), but I had some "blockers" with pricing strategies and I'm not a PR person so the product gained very little traction. It's offline now due to my other commitments but I'm keen to look into re-starting it again if there's enough interest.
I don't want to use their CI. I already have CI for running tests and other validations. I don't want to have a separate single-threaded CI stack with bare minimum features for building the site.
These all have "atomic" deploys which cause problems with hash-named assets as you get 404s during the transition.
What I basically want is something that I can rsync to and that is served by a CDN. I've ended up with S3+CloudFront but it doesn't have all of the nice features you may want from a static site host like index pages, but that is pretty easy to work around.