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.