$base_url = ''; in settings.php should be possible without notice warnings | Drupal 7.65

I do not think this is a bug. Documentation of the $base_path variable in default.settings.php states:

/**
* Base URL (optional).
*
* If Drupal is generating incorrect URLs on your site, which could
* be in HTML headers (links to CSS and JS files) or visible links on pages
* (such as in menus), uncomment the Base URL statement below (remove the
* leading hash sign) and fill in the absolute URL to your Drupal installation.
*
* You might also want to force users to use a given domain.
* See the .htaccess file for more information.
*
* Examples:
* $base_url = 'http://www.example.com';
* $base_url = 'http://www.example.com:8888';
* $base_url = 'http://www.example.com/drupal';
* $base_url = 'https://www.example.com:8888/drupal';
*
* It is not allowed to have a trailing slash; Drupal will add it
* for you.
*/
# $base_url = 'http://www.example.com'; // NO trailing slash!

See: ...fill in the absolute URL to your Drupal installation.... Empty string is not a valid value for this variable.