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As far as I can see, the GitHub documentation does not refer to a "primary" branch but it does refer to a "default" branch. This may seem like verbal nit-picking but I think it's an important distinction. The default branch isn't necessarily "more important" than other branches, it is just the branch that is used by default in a couple of places:

 > The default branch is the branch that GitHub displays when anyone visits your repository. The default branch is also the initial branch that Git checks out locally when someone clones the repository. Unless you specify a different branch, the default branch in a repository is the base branch for new pull requests and code commits.

As you correctly mention in your own answer, the default branch is used to determine what is shown by default in the web interface when someone visits your repo. Presumably it would be quite possible for the GitHub developers to implement additional UI to require visitors to explicitly choose a branch rather than just showing the default, but this would be extra development work for an unclear benefit.

The second usage ("the initial branch that Git checks out locally when someone clones the repository") is arguably far more important, because it is not within the control of GitHub developers. The Git tool requires a "currently active branch" in the remote repo, because this is what the `git clone` command checks out after cloning:

> Clones a repository into a newly created directory, creates        remote-tracking branches for each branch in the cloned repository        (visible using git branch --remotes), and creates and checks out an initial branch that is forked from the cloned repository’s currently active branch.

Since the Git tool needs a currently active branch, GitHub needs some mechanism to decide what this branch should be, and the current logic they are using is "first branch in the repo, unless the repo owner chooses a different one". Having no default branch at all is not really an option unless either (1) nobody is ever expected to clone the repo using the `git` command, or (2) the upstream Git developers can be persuaded that there is a use case for a repo with no active branch, and they agree to add handling for this case (e.g. by interactively prompting the user which branch to check out after cloning).