Let's say I integrate with a CLI utility, which comes up. Docker wouldn't work? If the answer is no I think I'd say I understand the theory better. Unless docker is providing some kind of universal kernel API or something?
This does mean that you end up with multiple copies of the same utilities and libraries on the system - although docker does have a mechanism to avoid duplication between different docker containers. It also means that if you take your docker container and move it over to a different system with a newer operating system (or a different distribution) all the parts of the operating system it cares about come with it.
The Linux kernel - the part that gets shared between containers - is mostly just responsible for process and memory management, networking and interacting with the computer hardware.