How should I use starter content utility to show some default menus?

I think you’ve a typo here:

'nav_menu'  =>  array(

Replace it with:

'nav_menus'  =>  array(

We can also check out how it’s setup in the default Twenty Seventeen theme and look further how it’s implemented within get_theme_starter_content():

// And nav menu items are grouped into nav menus.
case 'nav_menus' :
    foreach ( $config[ $type ] as $nav_menu_location => $nav_menu ) {

        // Ensure nav menus get a name.
        if ( empty( $nav_menu['name'] ) ) {
            $nav_menu['name'] = $nav_menu_location;
        }

        $content[ $type ][ $nav_menu_location ]['name'] = $nav_menu['name'];

        foreach ( $nav_menu['items'] as $id => $nav_menu_item ) {
            if ( is_array( $nav_menu_item ) ) {

                // Item extends core content.
                if ( ! empty( $core_content[ $type ][ $id ] ) ) {
                    $nav_menu_item = array_merge( $core_content[ $type ][ $id ], $nav_menu_item );
                }

                $content[ $type ][ $nav_menu_location ]['items'][] = $nav_menu_item;
            } elseif ( is_string( $nav_menu_item ) && ! empty( $core_content[ $type ] ) && ! empty( $core_content[ $type ][ $nav_menu_item ] ) ) {
                $content[ $type ][ $nav_menu_location ]['items'][] = $core_content[ $type ][ $nav_menu_item ];
            }
        }
    }
    break;