You may notice this permission error "You do not have sufficient privileges to perform this action" when previewing an email even though the user role has permission to access the emails.
This happens when the user role didn't have any access to the lead database as shown in the picture.
The user role needs the top-level access to the Lead Database(1 of 15). This is because the email previewer allows one to view the asset as if you were a specific lead, but if you can't see the leads, then the error will trigger.
To fix this issue, you would need to provide the user role with top-level access to the lead database.