Running your static site behind a server certificate on s3 is easy'ish, but very confusing if you don't know how to do it. To start you need to connect s3 to cloudfront to certificate manager then to your dns. It is easier than it sounds.

Automating your deployment can be one of those things that is tricky to get right at first, but pays off in spades once done. It seems as you develop you continually evolve your processes until you finally figure out how to get your deploys...