It's very important to remember that on your ArcGIS Server machine, it is the ArcGIS Server Account that will be trying to access the Oracle Client files. This user may have lower permissions than the user you were logged in as when you installed the Client, or it may not have access to a User level PATH Environment Variable.
Check the ArcGIS Server Account has full access to the Oracle Client folders.
Check your PATH Environment Variable is pointing to the Oracle Client for the ArcGIS Server Account, or is set at System level.
Alternatively, log in to the ArcGIS Server machine as the ArcGIS Server Account user to check Permissions or PATH settings.
https://enterprise.arcgis.com/en/server/latest/deploy/windows/accounts-used-by-arcgis-server.htm