Some installations might wish for a phame blog to be public on a custom domain without making the entire installation public.
Currently this is not possible, and as of D8918 setting as custom domain on a non-public installion is no longer possible either.
It would be a nice feature to have custom domain support without having to open up everything.
My two cents here are:
Sites eventually got split out (T5702) so that's no longer an issue, but this still isn't really straightforward. We do a lot of object loads which need some real viewer (blog, post, other adjacent posts, author, subscribers, object references in remarkup).
We can't use an omnipotent viewer because then you can violate policies on objects by mentioning them in a blog post.
We can't use a public viewer because they can't see anything.
We don't have a "generic logged-in user without any actual identity" user, since there are no use cases for it. And we can't let you pick a user to act as because then you can pick alincoln, mention/embed objects in a post, and violate policies.
I'm not sure we really have much in the way of use cases for this, and Phame blogs work better now as internal documents than they did in 2014, so maybe that satisfies things. I don't think we need to pursue this for unbeta. The UI is reasonably helpful about warning you about this upfront.