Field rights (such as the ability to view or edit the contents of the field) are specifically tied to the field themselevs. The administration console lets you edit them directly from the template, but they aren't actually properties of the template. If you simply want to ensure that only HR and Admin sees the template, and don't care about the fields inside of the template, you can simply ensure that they are the only trustees that have the 'read' template access right allowed (the one in the middle section), and then leave the field access rights alone. That way only the admin or a member of that group will be able to see or select this template, but all users could see values assigned to these fields, regardless of template state. 

Note that you might want to also restrict other users being able to view the values of one or more of these fields (for example I see SSN there), and that you'd do in the bottom section.