Custom taxonomy archive not showing posts of certain child taxonomies

What you are calling “filter” is just a HTML select element. You need the put the select element in a form and filter the posts based on the selected option of this select element.

For example, for the frontend:

The form:

<form method="post" action="<?php echo get_post_type_archive_link('rlt_rule');?>">
    <?php
    $filter_taxonomy = 'rlt_rule_headline';
    wp_dropdown_categories( 
        array(
            'show_option_all' => 'Show all headlines',
            'taxonomy'        => $filter_taxonomy,
            'name'            => $filter_taxonomy,
            'orderby'         => 'name',
            'selected'        => ( isset( $wp_query->query[$filter_taxonomy] ) ? $wp_query->query[$filter_taxonomy] : '' ),
            'hierarchical'    => true,
            'depth'           => 3,
            'show_count'      => false,
            'hide_empty'      => true,
        ) 
    );
    <button type="submit">Search</button>
</form>

The filter process:

add_action( 'pre_get_posts', 'my_pre_get_post' );
function my_pre_get_post($query){

   //limit to main query and frontend are. Add you own conditionals
   if($query->is_main_query() && !is_admin()) {

       $tax_query = array();
       $tax_input = isset($_POST['rlt_rule_headline']) ? $_POST['rlt_rule_headline'] : '';
       if(!empty($tax_input)){

           $tax_query[] = array(
                                'taxonomy'         => 'rlt_rule_headline',
                                'field'            => 'id',
                                'terms'            =>  $tax_input,
                                'operator'         => 'IN',
                                'include_children' => true
                               );
        }
    }

    $query->set('tax_query',$tax_query);
}

For the backend you to include your filter in the restrict_manage_posts action hook as seen in this answer.