I don’t see “display in web site” anywhere on the Presentation tab. I only recall seeing that checkbox on Item records.
I’m actually not the admin but the custom admin-like role assigned to me has “Set Up Web Site” role assigned to it. It’s the only role that has this permission and I’m the only user with this role. I’m able to see the Presentation tab in our intranet when I’m signed in to any of the more basic custom roles (shipping, sales, etc) assigned to me.