I would like to allow some users to download a zip archive of specific tags/commits, but not give them the ability to clone or browse the whole repository.
That is not possible. The best approach to this one would be creating a new repository with the history you want to share. When needing to update the new repository you can create pull requests and update it. This would be the public-facing repository where you would allow people to clone it.
You probably wanted something simpler, but I can think of any better option now.
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