I host all my APIs on Heroku and all my static front ends on S3 with CloudFront on top. No downsides in my mind. Free and instant TLS certs with CloudFront, too.
My search led me to Netlify, which seems pretty great. It runs your build script when you push to GitHub, then serves the resulting static files with a Let's Encrypt certificate. Comes built-in with a CDN, so you don't need CloudFront or CloudFlare.