Pods created after the first try to load the private files using the same path as the first one
Posting my comment as an answer.

sites/default/files needs to be on a shared file system so that Drupal can work correctly. This contains uploaded and generated files (e.g. CSS/JS).

If you do this, you probably want to move twig to a local temporary folder, see Can I change the location of the cached Twig templates?. Note that you then need to take care of emptying that folder on all servers when you do a deployment (Unless you do something really special, this cache only needs to be updated when you change your template files).