We usually use CloudFront and S3 for static HTML sites. Lambda could be useful to make dynamic.

A flat website doesn't seem to have much need for it..?

At work for simple static websites we use S3 and, you guessed it, Cloudfront.

We tend to use Cloudfront and S3 for static too, but we've used APIG + Lambda for generating dynamic share pages in some apps, either just for bots or with some simple styling