bavard unfortunately, not really. I think you'll have to mark every primary tag as restricted, and configure the permission on each of these tags.
If you have few primary tags this shouldn't bee too much of a concern, but I understand this can get tedious if you have many private primary tags.
If needed you could get an extension developed to make the opposite, choose a tag where global permissions don't apply.
If the introduction post does not need to be public, maybe this extension can help ? https://discuss.flarum.org/d/23004-flarum-help-tags So you can hide everything from public view, but let new members create a discussion only them and existing authorized users can see.
