Who sees what
Appearance > Axle Page Template Usage needs the Axle Page Template Usage capability (edit_pages by default). On top of that, every screen and every ability only covers the post types the user can edit (edit_posts on the post type): the counts, the Move buttons, the theme check, the history, the undo and the "Changed by" filter. Someone who can edit pages but not posts sees nothing about posts. In the History tab, a page the user may not read (a draft or private page of someone else) shows its ID instead of its title.
Checking a theme lists the installed themes, so the "Check a theme" tab, its link on Appearance > Themes, the notice after a switch and the check-theme ability also need the theme check capability (switch_themes by default). A saved plan only covers the post types the person who saved it can edit.
WP-CLI has no user, so it covers every post type.