No confirmation message after saving primary links

You are correct, there is no message presented that indicates the links were saved. The only visual feedback is the *s created when reordering the menu items will disappear once saved. This is common on most of the draggable tables like this in Drupal.

It should be easy to throw a dsm to the user indicating the save, but this would probably need to be something Drupal wide, rather then just the menu page. It looks like this is a similar issue that has been fixed in D7 #902738: No status message when reordering fields. and may fix this problem too.

Here is a patch for menu module. It adds an alert stating the action once the menu configuration is saved.

And here is a snippet of code which does the job from a custom module (change 'mymodule' by any custom module name where this fits):

/**
 * Implementation of hook_form_alter()
 */
function mymodule_form_alter(&$form, $form_state, $form_id) {
  switch ($form_id) {
    case 'menu_overview_form':
      $form['#submit'][] = 'mymodule_add_post_save_alert_form_submit';
  }
}

/**
 * Submit handler that adds an alert after saving menu configuration
 */
function mymodule_add_post_save_alert_form_submit($form, &$form_state) {
  drupal_set_message(t('The menu configuration was saved.'));
}

I do not know why did the patch failed testing.

Changing status to "needs review".

Doh!