During a twitter conversation with itzg, he suggested that I use [docker's built in --restart="unless-stopped"
runtime flag] (https://gist.github.com/itzg/36e59186f6195a445f1f1fa2b0087ea9). That worked a treat and means I don't need to maintain an extra service.
This is only retained for historic record. Don't follow these steps.
If you want to run a minecraft server in a docker container on Ubuntu 16.04 and have it automatically start at boot, follow these steps: