This is not something that is unique to docker though. I could just as easily set up a hook to deploy our site without docker, but it's not something I need.
I prefer to manually deploy, it's only one command, and I can make sure it's all worked correctly.
That said, I will be moving that command to a chat command, as I like that. But even then, it'll be a manually triggered command.
I have auto deploy setup for CI testing, as in that case I do want to know that those branches are ready for deployment to prod, when I want to.