How to create and populate with few links a menu in child theme functions.php?

Question

Here you go.
Since you already have the menu-id all you need is:

$menu_item_defaults = array(
                    'menu-item-db-id' => $menu_item_db_id,
                    'menu-item-object-id' => 0,
                    'menu-item-object' => '',
                    'menu-item-parent-id' => 0,
                    'menu-item-position' => 1,
                    'menu-item-title' => 'Play Game',
                    'menu-item-url' => '',//not sure is a blank is ok here
                    'menu-item-description' => '',
                    'menu-item-attr-title' => '',
                    'menu-item-target' => '',
                    'menu-item-classes' => '',
                    'menu-item-xfn' => '',
                    'menu-item-status' => 'publish',
                );

wp_update_nav_menu_item('primary-menu', 0, $menu_item_defaults);

However, it you’re trying to speed deployment of a site with known content try it this way. You can create the page and add it to the menu all on the fly.

In my case I’m creating a custom menu first but you can use this to get the id of any menu.

$menu_exists = wp_get_nav_menu_object( 'Public Menu' );
if ( $menu_exists === false ) {
    $var_pub_menu_id = wp_create_nav_menu( 'Public Menu' );
} else {
    $var_pub_menu_id = $menu_exists->term_id;
}

Then I create my page and add it to the menu:

$default_fgpsn_page = array(
    'post_title'    => $fgpsn_default_pages[$j],//any page title. I'm working in a loop here
    'post_status'   => 'publish',
    'post_type'   => 'page',
    'post_content'   => 'Property Browse Short Code',//this can even be a shortcode
    'post_author'   => 1
    );

   $default_fgpsn_page_id = wp_insert_post( $default_fgpsn_page );

            $menu_item_defaults = array(
                //'menu-item-db-id' => $menu_item_db_id,
                //'menu-item-object-id' => 0,
                //'menu-item-object' => '',
                'menu-item-parent-id' => 0,
                'menu-item-position' => 1,
                'menu-item-title' => $fgpsn_default_pages[$j],
                'menu-item-url' => get_page_link( $default_fgpsn_page_id ),
                'menu-item-description' => 'public properties listing',
                'menu-item-attr-title' => '',
                'menu-item-target' => '',
                'menu-item-classes' => '',
                'menu-item-xfn' => '',
                'menu-item-status' => 'publish',
            );

            wp_update_nav_menu_item($var_pub_menu_id, 0, $menu_item_defaults);

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