By default, the edit_others_posts permission is only given to Super Admins, Administrators and Editors.
– Who has the edit_others_posts permission?
Is the author dropdown missing for all user roles, i.e., contributor, author, editor, and administrator?
Are you using a custom post type?
Are you using a custom role and/or any plugins that manage user roles?
Try creating a new user for testing purposes. Assign this user the role of editor. Check if this results in the author dropdown reappearing.
Try installing the User Role Editor plugin and check if the edit_others_posts permission is enabled for whatever role(s) you’re encountering the problem with.
A plugin or theme might be the problem. Try disabling your plugins one by one. Try using Twenty Twenty-Two theme.
