Auto-realod with new posts // Ajax call

If what you want is is a full page refresh then you don’t need ajax at all and inserting the htt-equip html meta will do the trick

<meta http-equiv="refresh" content="5" >

(content is time in seconds IIRC)

You just hook on wp_head and insert it.

function wpse_228671_wp_head() {
  echo '<meta http-equiv="refresh" content="5" >';
}

add_action('wp_head','wpse_228671_wp_head');