I noticed that the Piklist Demos content type uses custom post statuses.

  ,'post_states' => true
      ,'status' => array(
        'draft' => array(
          'label' => 'New'
          ,'public' => true
        )
        ,'demo' => array(
          'label' => 'Demo'
          ,'public' => true
        )
        ,'lock' => array(
          'label' => 'Lock'
          ,'public' => true
        )
      )

Two issues here:
1. when viewing all posts in the admin, they all show a status of “Draft” regardless of the chosen status type.

2. Permalinks don’t work, even if I flush rewrites.

Once I removed the above snippet everything worked fine.

I also added a snippet in my CPT plugin to flush rewrite rules.

  // Flush rewrite rules for custom post types
add_action( 'after_switch_theme', 'do_flush_rewrite_rules' );

// Flush your rewrite rules
function do_flush_rewrite_rules() {
  flush_rewrite_rules();
}