Change auditor does not register events when running powershell scripts on servers

I would look at it this way. The story is complex and we need to consider the whole picture with following points of control:

#1. Right to execute the script/ACL/permissions layer: Local Admin, OS rights (Logon as Batch Job, As Service) - controlled by GPO

#2. Log the execution actions on Server: Events (Logon as Batch Job, As Service), TS Logon, Interactive Logon. 

#2.1 Log access on Resources: File Access on Share by the script, other servers$ shares etc...

I would look at it this way. The story is complex and we need to consider the whole picture with following points of control:

#1. Right to execute the script/ACL/permissions layer: Local Admin, OS rights (Logon as Batch Job, As Service) - controlled by GPO

#2. Log the execution actions on Server: Events (Logon as Batch Job, As Service), TS Logon, Interactive Logon. 

#2.1 Log access on Resources: File Access on Share by the script, other servers$ shares etc...