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This is a feature request I'd love to see implemented at some point -- having an API for managed servers to perform basic functions like start/stop/reboot a server.
My main scenario would be for server monitoring in case a restart is needed; rather than relying on the built in service that checks every 5 minutes for the server being offline, I can roll my own monitoring solution to kick a server back to life after 1-2 minutes. While it's a small detail, short downtime vs 5-10 minutes downtime can make a big difference in players rejoining a crashed server
As an added bonus for some encouragement here's what you'll look like after implementing the support for our use: