It appears that all staff seem to have access to the editor for custom global menu sections, although if anyone but admin tries to save changes, a permission error is generated. Only admins should be able to see the pencil editor buttons and access the custom menu sections modals (not all staff). Also, only admins should see the globe icon that indicates a public menu section. Mods can see both the globe and pencil icons.
Thanks for the report @Lilly! The UI did indeed not reflect the fact that a moderator cannot create, edit, or delete a public category. This has been solved in the fix below: