$messages isn't an array of messages; it's the rendered output theme_status_messages(). From the point at which you're looking to add messages (template_process_page()), Drupal has already processed all the messages it's going to render.

If you're looking to print some debugging messages you can do something like this:

function MYTHEME_process_page(&$variables) {
  drupal_set_message('new test message');

  //This will overwrite any previous messages that were to be rendered
  $variables['messages'] = theme_status_messages(array('display' => null));
}

$messages isn't an array of messages; it's the rendered output of theme_status_messages(). From the point at which you're looking to add messages (template_process_page()), Drupal has already processed all the messages it's going to render.

If you're looking to print some debugging messages you can do something like this:

function MYTHEME_process_page(&$variables) {
  drupal_set_message('new test message');

  //This will overwrite any previous messages that were to be rendered
  $variables['messages'] = theme_status_messages(array('display' => null));
}