hi, this may be too late for you but i had exactly the same problem, and the way my theme must be set up, is if a page is set as public, but a sub-page of another page (set the page parent at the bottom of the page when editing or creating new pages) then it is available to be viewed by anyone that is not logged in, but does not appear in the menu. View my site https://www.lewisurwin.com and click on portfolio for an example.
I hope this has been of some help to you
Lewis Urwin
