How can I get WP to use templates in lower-level subfolder?

Question

Whilst you’re correct in saying WP will only scan directories 1 level deep for templates, since 4.4 you have complete control over the theme_page_templates filter (you can now add to as well as remove from the list):

add_filter( 'theme_page_templates', function( $templates ) {
     $templates['page-templates/another/directory/template.php'] = 'Another Directory Template';

     return $templates;
});