Within JN, you cannot edit/save in the built-in report system unless you have allowed access to Account Settings and Built-in Reports, or created the report as that user.

I need staff (admins, supervisors, and other team members) to be able to edit reports created by superiors, but I have ZERO desire for them to have Account Settings access.

There is an ability to give someone access to the built-in report system. Why would there also be another access for them to be able to edit/save reports, especially when that is tied to the entire Account Settings?