How to check if page exists using page alias in D7?
If you look at the code for drupal_valid_path() it passes the alias through url_is_external(), which considers paths with a http:// prefix to be external. The function defaults to allow access to these sorts of paths and simply returns that fact; it does not check whether the 'external' url exists. 

Since you're adding $GLOBALS['base_url'] to the alias it will appear to be external; you just need to pass the alias into the function without the base URL attached:

if(drupal_valid_path('weird_alias')) {
  // This code is never run
  print_r('Exists!');

}