The reorder screen checks a page capability, not a posts capability. I have watched Authors hunt for a menu they will never see. Editors sail through. The check is edit_pages on the menu and again on the save request.
Who can do what
| Account | Reorder screen | Settings |
|---|---|---|
| Administrator | Yes | Yes |
| Editor | Yes. Editors have edit_pages | No. Settings need manage_options |
| Author | No | No |
| Contributor, Subscriber | No | No |
An Author can publish their own posts and still cannot drag them here. Giving Authors edit_pages would also let them edit other people’s pages, so I would not flip that capability just to reveal this menu. An Editor, or a custom role that already has edit_pages, is the fit.
The save request
Dragging sends an admin-ajax request named post_sort. It checks edit_pages and a nonce. A failed check returns an empty response and the row will not stay where you dropped it. If a drag snaps back, look at the account before you look at the list.
Items per Page is per user. One Editor can work with 50 rows and another with 100. It does not change who is allowed in.