Earlier quoted context omitted.
Aptly is a repository manager, it works in terms of tightly defined package versions and dependencies. Docker is for deployment, it works in terms of graphs of filesystem images. The tools do slightly overlap in use case, but they operate in entirely different domains. It wouldn't be unreasonable to mix both in the same project, but there is no hard rule for when this would be appropriate.
I think I will stick with a Docker. DEB packages are very complicated and this tool does not really handle dependency conflits and so on that well. Plus it does not work with RPM and other packages systems.
I'm not going to be building .debs for installation inside Docker containers (turtles all the way down) but I would still consider something like aptly for storing and managing any obscure dependencies that my applications might require, and for maintaining a mirror on the local network to speed up image builds.