Is Docker right for my use case?
I'm not 100% sure I understand from the question, but it sounds like the Docker solution would be to go from having an (physical?) appliance with an OS and your app installed on it, to having an appliance with an OS and Docker on it, running a single container with your app in it. That doesn't obviate the need to update the OS in the host, and it adds a layer of complexity (and more updates to contend with, as you'll now have to keep Docker and the OS patched) with no readily-apparent benefit as far as the specific areas mentioned in the question are concerned.

However, if you're talking about going from a virtual appliance to a Docker container, that could potentially smooth things out for you, but it also adds Docker as a dependency for your product; you're shutting out anyone who isn't using Docker and doesn't want to add it to their stack just to use your product. You could continue to support those that don't/won't use Docker by continuing to ship the (now "legacy") virtual appliance as before, but now you've just doubled your workload because you have two distributions to support instead of one.