How would I add the comment count to this sub-pages of current page snippet

A custom walker can be passed to wp_list_pages() using the $walker parameter.

Here is a custom walker named WPSE_Walker_Page_Comment_Count() which is based on the default page walker, Walker_Page(). We’re only overriding the start_el() method in this case. The output has been customized to include the comment count.

First, add the walker code to your theme’s functions.php or to a plugin:

class WPSE_Walker_Page_Comment_Count extends Walker_Page {
    // Copied from Walker_Page::start_el() core version v4.7.4.
    /**
     * Outputs the beginning of the current element in the tree.
     *
     * @see Walker::start_el()
     * @since 2.1.0
     * @access public
     *
     * @param string  $output       Used to append additional content. Passed by reference.
     * @param WP_Post $page         Page data object.
     * @param int     $depth        Optional. Depth of page. Used for padding. Default 0.
     * @param array   $args         Optional. Array of arguments. Default empty array.
     * @param int     $current_page Optional. Page ID. Default 0.
     */
    public function start_el( &$output, $page, $depth = 0, $args = array(), $current_page = 0 ) {
        if ( isset( $args['item_spacing'] ) && 'preserve' === $args['item_spacing'] ) {
            $t = "\t";
            $n = "\n";
        } else {
            $t="";
            $n = '';
        }
        if ( $depth ) {
            $indent = str_repeat( $t, $depth );
        } else {
            $indent="";
        }

        $css_class = array( 'page_item', 'page-item-' . $page->ID );

        if ( isset( $args['pages_with_children'][ $page->ID ] ) ) {
            $css_class[] = 'page_item_has_children';
        }

        if ( ! empty( $current_page ) ) {
            $_current_page = get_post( $current_page );
            if ( $_current_page && in_array( $page->ID, $_current_page->ancestors ) ) {
                $css_class[] = 'current_page_ancestor';
            }
            if ( $page->ID == $current_page ) {
                $css_class[] = 'current_page_item';
            } elseif ( $_current_page && $page->ID == $_current_page->post_parent ) {
                $css_class[] = 'current_page_parent';
            }
        } elseif ( $page->ID == get_option('page_for_posts') ) {
            $css_class[] = 'current_page_parent';
        }

        /**
         * Filters the list of CSS classes to include with each page item in the list.
         *
         * @since 2.8.0
         *
         * @see wp_list_pages()
         *
         * @param array   $css_class    An array of CSS classes to be applied
         *                              to each list item.
         * @param WP_Post $page         Page data object.
         * @param int     $depth        Depth of page, used for padding.
         * @param array   $args         An array of arguments.
         * @param int     $current_page ID of the current page.
         */
        $css_classes = implode( ' ', apply_filters( 'page_css_class', $css_class, $page, $depth, $args, $current_page ) );

        if ( '' === $page->post_title ) {
            /* translators: %d: ID of a post */
            $page->post_title = sprintf( __( '#%d (no title)' ), $page->ID );
        }

        $args['link_before'] = empty( $args['link_before'] ) ? '' : $args['link_before'];
        $args['link_after'] = empty( $args['link_after'] ) ? '' : $args['link_after'];

        // Get the comment count for the page.
        $comments_number = get_comments_number( $page->ID );
        if ( 0 == $comments_number ) {
            $comments_number_output = __( ' — No Repsonses', 'text_domain' );
        } else {
            $comments_number_output = sprintf(
                /* translators: number of comments */
                _nx(
                    ' — %1$s Response',
                    ' — %1$s Responses',
                    $comments_number,
                    'comments title',
                    'text_domain'
                ),
                number_format_i18n( $comments_number )
            );
        }
        $output .= $indent . sprintf(
            '<li class="https://wordpress.stackexchange.com/questions/267058/%s"><a href="https://wordpress.stackexchange.com/questions/267058/%s">%s%s%s%s</a>',
            $css_classes,
            get_permalink( $page->ID ),
            $args['link_before'],
            /** This filter is documented in wp-includes/post-template.php */
            apply_filters( 'the_title', $page->post_title, $page->ID ),
            $args['link_after'],
            sprintf( '<span class="comment-count">%s</span>', // Append comment count.
                $comments_number_output
            )
        );

        if ( ! empty( $args['show_date'] ) ) {
            if ( 'modified' == $args['show_date'] ) {
                $time = $page->post_modified;
            } else {
                $time = $page->post_date;
            }

            $date_format = empty( $args['date_format'] ) ? '' : $args['date_format'];
            $output .= " " . mysql2date( $date_format, $time );
        }
    }
}

Then in your template, call wp_list_pages() like this:

<ul><?php
    wp_list_pages( array(
        'title_li'    => '',
        'depth'       => 1,
        'child_of'    => get_the_ID(),
        'sort_column' => 'post_title',
        'show_date'   => 'modified',
        'date_format' => $date_format,
        'walker'      => new WPSE_Walker_Page_Comment_Count(),
    ) );
?></ul>

Note that I used get_the_ID(), which is cleaner than using the global $id variable. Also, $date_format was never defined in your original code. The most important part is the addition of the $walker parameter, which tells wp_list_pages() to use our custom walker to generate output.

Example output (formatted for legibility):

<ul>
    <li class="page_item page-item-1090">
        <a href="http://example.com/parent-page/child-page-01/">Child Page 01<span class="comment-count">&nbsp;— 1 Response</span></a>
    </li>
    <li class="page_item page-item-1092">
        <a href="http://example.com/parent-page/child-page-02/">Child Page 02<span class="comment-count">&nbsp;— No Responses</span></a>
    </li>
    <li class="page_item page-item-1094 page_item_has_children">
        <a href="http://example.com/parent-page/child-page-03/">Child Page 03<span class="comment-count">&nbsp;— 2 Responses</span></a>
    </li>
    <li class="page_item page-item-1096">
        <a href="http://example.com/parent-page/child-page-04/">Child Page 04<span class="comment-count">&nbsp;— No Responses</span></a>
    </li>
    <li class="page_item page-item-1098">
        <a href="http://example.com/parent-page/child-page-05/">Child Page 05<span class="comment-count">&nbsp;— No Responses</span></a>
    </li>
</ul>