Keep in mind that mysqli::reap_async_query only returns mysqli_result on queries like SELECT. For queries where you may be interested in things like affected_rows or insert_id, you can't work off of the result of mysqli::reap_async_query as the example in mysqli::poll leads you to believe. For INSERT/UPDATE/DELETE queries, the data corresponding to the query can be accessed through the associated key to the first array in the mysqli::poll function.

So instead of
<?php
    foreach ($links as $link) {
        if ($result = $link->reap_async_query()) {
            print_r($result->fetch_row());
            mysqli_free_result($result);
            $processed++;
        }
    }
?>

The data is accessible via:
<?php
    foreach ($links as $link) {
        if ($result = $link->reap_async_query()) {
            //This works for SELECT
            if(is_object($result)){
                print_r($result->fetch_row());
                mysqli_free_result($result);
            }
            //This works for INSERT/UPDATE/DELETE
            else {
                print_r($link);
            }
            $processed++;
        }
    }
?>