Insufficient access right to user error while accessing email attachments in Salesforce

I am facing Insufficient access right to user error while trying to access email attachments through programmatic backend code. This is working fine for System Administrator group however, for any other profiles we are getting this error despite giving those profiles full access to Case object. Other profiles are able to access cases and case attachments but they are NOT able to access email attachments attached to cases.

It seems Salesforce treats email attachments different than case attachments and accessing email attachments through programmatic backend code may require additional or specific permission. As a workaround, we followed below mentioned steps to give “MODIFY ALL DATA” permission to intended profile. However, this is NOT ideal solution and hence wanted to check why Salesforce is treating Case Attachments and Email Attachments on Cases differently and what’s the best way to resolve this issue.

– click on SET-UP link
– search PROFILE in QUICK-SEARCH box
– go to the profile which you want to give permission.
– click on EDIT button.
– go to Administrative Permissions section and select MODIFY ALL DATA checkbox
– now click on SAVE button