My bet that you'd migrate to GRPC eventually as you scale :) I like the simplicity of HTTPS/JSON API as well, but it just broke down for us at a certain scale point.
Re: Teleport with EKS
True, CNCF clusters support mTLS out of the box, but EKS hides the endpoint and does not let you provision CA to trust. You will have to run teleport proxy inside the EKS cluster to translate mTLS to EKS IAM auth. However, you don't have to have a centralized proxy, you can just deploy Teleport proxy agent in each cluster and hide your K8s endpoint.
You also don't have to have a single Teleport proxy to do that.