If we plan to use pool_hba.conf, then we must use the pool_passwd file to maintain passwords for all database users. This approach requires that every time a new user is created, their password is added to the pool_passwd file. The challenge with this method is that it becomes difficult to maintain, as we need to manually update the pool_passwd file for every user creation or password change. Additionally, after updating the pool_passwd file, Pgpool needs to be restarted for the changes to take effect.
This behavior might be related to how Pgpool handles authentication.
Having never used pgpool I very well could be wrong but from what I can gather what you want to do is simply not possible within the current design of pgpool. Namely you have multiple nodes so cannot rely on raw mode but raw mode is required for the kind of pass-through auth you are trying. Instead you require pool_hba.conf which itself requires the password file.