Below I have changed the default tab at `/admin` to `/admin/index`, but now `/admin/tasks` is 403.  Even if I change `$items['admin/tasks']['access callback']` to `TRUE` I'm still denied access.

    function hook_menu_alter(&$items) {
      $items['admin/tasks']['page callback'] = 'system_admin_menu_block_page';
      $items['admin/tasks']['access arguments'] = $items['admin']['access arguments'];
      $items['admin/tasks']['file'] = $items['admin']['file'];
      $items['admin/tasks']['type'] = MENU_LOCAL_TASK;

      $items['admin']['page callback'] = $items['admin/index']['page callback'];

      $items['admin/index']['type'] = MENU_DEFAULT_LOCAL_TASK;
      unset($items['admin/index']['page callback']);
      unset($items['admin/index']['access arguments']);
      unset($items['admin/index']['file']);
    }

How can I both restore access to `/admin/tasks` and set the default tab to `/admin/index`?

Resolution: The admin module was the problem.  It's enabled by default in my installation profile.