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?

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.