The image acts as a foundation for each container that you start from it.
You might do the update in your 'fashion' way, but containers are ephemeral - let's say it got restarted, you updates are lost and you are back to the previous version.
If you want to handle this process by yourself, how would you track your versions ? Essentially you would have a root docker image and after that it would be so hard to know which version is currently running ? Could you also safely roll-back version to version ? How about if somebody hijacks you container, due to some Unix Socket permissions with the Docker daemon ...
What I am trying to say, do not rediscover the wheel :)
Take advantage of modern technologies like Kubernetes, deploy each image for a new version there, you have a roll - out deployment, readiness probes and so on ... You would be safe that your software is always working, track the versions and have H/A.
