find Menu Settings of node, given URL alias (or node ID)
Comments in my question getting URL from Route: breaks in localhost subdirectory have clarified that the solution of @4k4 is absolutely the right way to get the alias from the current path.  But the alias is not as useful as I thought in applying suggestion (@Ivan Jaros) to use entityQuery to take it the rest of the way, since the current path is able to match the link.uri field of menu_link_content, except for a leading slash in the pathname and the entity: protocol:

$current_path = \Drupal::service('path.current')-&gt;getPath();
$system_uri = 'entity:' . trim($current_path, '/');

$query = \Drupal::entityQuery('menu_link_content')
  -&gt;condition('link.uri', $system_uri)
  -&gt;condition('menu_name', 'main');

$result = $query-&gt;execute();
$menu_link_id = (!empty($result)) ? reset($result) : FALSE;

if ($menu_link_id) {
  $menu_link = MenuLinkContent::load($menu_link_id);
  $menu_link_title = $menu_link-&gt;getTitle();
} else {
  drupal_set_message(t('no Menu Link Content for ID that we found...'), 'error');
}