Is there any advantage to this over something like cloudflare pages?
In terms of advantages, I'd say price, and the maturity of using AWS. I deploy a static site with a GitHub Action (look at my other repo whisky-menu for how to do this) which builds a React app, copies it to S3, and invalidates the CloudFront cache.
Because I have other resources and projects in AWS I'll keep this setup. It's so close to free that cost doesn't bother me.