It is possible to set up a menu item with hook_menu that points to an existing path?
You can only use hook_menu() if you're defining new paths, ideally, paths that are provided by the module implementing the hook.

If you want the Delete link to show up as a tab, hook_menu_alter is definitely the way to go, and adjusting the context value is what does the magic - as per Step 2 from marcvangend's post. If you want to change the "access callback" as per your snippet, then you can do that in hook_menu_alter() too, although I'm not sure why you'd want to let anyone delete pages?