Remove Docker-in-Docker from builds

Remove all traces of Docker-in-Docker from builds. The rationale is explained
generally in the `docs/builds.md` file. In short, Docker-in-Docker isn't secure
or stable for production, and there's no indication it will be in the forseeable
future. The codebase can be simplified by eliminating it, and our focus and
efforts can shift to solving the remaining issues with host Docker access.