Nested Page Template not showing in page attributes

In WP 3.4 they allowed adding template files one level deep.

Since then code has changed, however in wp-includes/class-wp-theme.php, in offsetGet() (from __get()) you’ll see:

case 'Template Files' :
    $files = $this->get_files( 'php', 1 );

Although I can’t fully unravel the maze of code for loading a theme, the above suggests the second argument, $depth, is still set to only 1 deep for Template Files. Which would coincide with the troubles your experiencing.

You may want to re-think your themes folder structure, and/or your logic for selecting a template.

If you’re conditionally checking which template was chosen based on a user selection in the first place, on your conditions using the load_template() andor template_redirect() might help out more than leaving it to WordPress defaults.