Just want to add here - Plesk does an excellent job with Git. It would be a great reference for a good implementation of this. I use it for all WordPress sites (custom themes and plugins). I think git is essential for hosting providers - anybody that is good at making websites will use git, either for static sites or WordPress components.
Something Plesk does really well is handle keys. Github only allows deploy keys to be used once. Adding multiple keys in cPanel is a pain. Plesk makes it super easy to generate multiple keys for different repos. Auto deploy and branch switching is easy. The other nice thing compared to cPanel is that the .git folder is not dumped into the web directory - deployment copies just the repo files over, not any of the git files.
One thing that would be cool to see here as well would be a way to automate deployment - either an API endpoint or some sort of git action. Currently, with Plesk we have to log in and pull latest.