I think there is a problem with overloading the term API gateway. That's really an orchestration service that might often live behind the gateway itself. In fact in some architectures each of those service calls would go through their own API gateway, if they serve both clients and services.
I agree with the idea and have used this architecture multiple times, but calling it an API gateway (which, to be fair, is exactly what I called it the first time I proposed it in design meetings) is going to confuse folks who are already working on cloud architecture.