With 1.10 you can just enable User namespaces, which allows for root in a container to map to a non-privileged user outside the container, that way it's a one-time (per instance) change.
With that said there's a couple of ways this is getting addressed.
1) in 1.10 authorization plugins landed as a feature,so it's possible to add this functionality. 2) there's a number of services which run on top of Docker Engine (e.g. Docker Universal Control Plane) which have authentication/authorisation at that level.
Nonetheless, you shouldn't need to run anything as root in order to start a container that doesn't require extra privileges .