Should I be doing something else?
Yes. ๐
Go to Administration > Modules > Tabs and review the access level you have set for each tab. While there you can also think about the display order of the tabs, in case you prefer to display them differently to the default order. I recommend that you also do the same for other items in that Administration > Modules group (Menus, sidebar modules, blocks (available on the home page), Charts, Lists, Reports)
The “access” level should be set to the lowest level of users who you think should be able to see that part of the system.