I must be missing something. Tried your first suggestion...
On the Access tab I have:
Allow new subscriptions Yes
Allow access from site frontend Yes
Changed visible for access to Super Users
Allow access from site frontend to Super Users
Result in the frontend is that this plan shows as an option as if it is Public.
If I log in as an admin super user, and go to the registration screen...I get
Error
You are already logged in
If I go to the user list and choose a member with this plan already
I see initially the members subscription. If I go to edit...the subscription tab is missing...
Hope you can straighten me out.