Ordering of gallery images without using shortcode in theme

You are correct.

The menu_order is no longer used for media in the gallery. Not sure if this is by design or an oversight. Could be by design since you can now include media in a gallery even if it is not “attached” to the page/post. In any case, below is the method I use to grab the ids and get the attachments based on the order specified in the shortcode:

// helper function to return first regex match
function get_match( $regex, $content ) {
    preg_match($regex, $content, $matches);
    return $matches[1];
} 

// Extract the shortcode arguments from the $page or $post
$shortcode_args = shortcode_parse_atts(get_match('/\[gallery\s(.*)\]/isU', $post->post_content));

// get the ids specified in the shortcode call
$ids = $shortcode_args["ids"];

// get the attachments specified in the "ids" shortcode argument
$attachments = get_posts(
    array(
        'include' => $ids, 
        'post_status' => 'inherit', 
        'post_type' => 'attachment', 
        'post_mime_type' => 'image', 
        'order' => 'menu_order ID', 
        'orderby' => 'post__in', //required to order results based on order specified the "include" param
    )
);

Hope that helps!