Loop posts without any taxonomy

One solution would be to give a value of “none” to your posts without any post group, similarly as you have “uncategorized” posts in WP Categories.

Alternatively… I haven’t checked it, but why don’t you try to make another WP_QUERY like this:

new WP_Query( array( 'post_type' => 'page', 'page_group' => '' ) )

error code: 523