Problem with Class, Filters and Callbacks

Here’s a slightly modified version of your code snippet:

add_action( 'login_head', [ 'WPSE_Admin', 'plugin_setup' ] );

class WPSE_Admin 
{
   public static function plugin_setup()
   {
      add_filter( 'login_headerurl', [ 'WPSE_Admin', 'the_logo_url' ] );   
   }

   public function the_logo_url()
   {
      return get_bloginfo('url');
   }
}

The filter callbacks must be public, not private. The reason for this is that apply_filters()/apply_filters_ref_array() are running call_user_func_array() on the stored filter callbacks, in the global $wp_filter array.

Also notice that you’re not instantiating your WPSE_Admin class, so you can’t use $this. You might want to use namespace but I just prefix the class here.