Cloudflare Pages looks like a more productionised version of this, yes.
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.
You're correct, yes, I didn't realise how generous the free tier is. If you're doing fewer than 500 updates per month, Cloudflare looks good https://pages.cloudflare.com/
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.
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.