I know you have git web hook and git deployment via a pull but I would love to see more functionality here. For example Envoyer.io and runcloud.io offer no downtime full pull with ability to run commands such as composer update. Also to be able to store say 6 versions of recent pulls.

You could have a releases folder and a current folder with a symlink to the latest release. This would enable the new website to only go live one full set of files and commands have been executed before updating symlink. Also enables you to go back to an older version incase of a mistake.