Current the ENVOY_ADMIN_API is used to prevent application startup until Istio/Envoy is ready. Istio itself has a readiness probe (used by k8s healthchecks) at http://127.0.0.1:15020/healthz/ready. Looking at the master branch of Istio this endpoint verifies the Istio Admin API is ready (or it will time out), and internally verifies Envoy is operational.
I think using Istio's readiness endpoint makes more sense as we can be more confident that not only Envoy is ready, but Istio itself is too. Most likely this would mean deprecating ENVOY_ADMIN_API as the name of the variable wouldn't make sense.