Yes, reproducibility requires source code for the entire dependency graph, down to a set of bootstrap binaries. This is what tools such as GNU Guix and Nix allow you to do. They enable reproducible science.
My original point is that the complexity of dealing with differing environments and code versions, combined with the sensitivity of floating point results on the exact source code and environment, practically kills the possibility for exact reproducibility.
That's why I'm claiming that a better goal is approximate reproducibility.
So a sufficient improvement in toolchains/package management etc. would probably enable bitwise reproducibility, but I think a lot of people in this thread, and the original article, were implying that current tools are sufficient and scientists just need to stop being lazy.