WHY Can't I Use SetTargetObjectId(UserId) and SetWhatId?

When I was faced with this, I

1. Had a fixed Contact record with firstname = null, lastname = ErrorNotificationsDoNotDelete, email = [email protected] under a known Account MyCompany *do not delete*
2. Used the the running User's email as the 'to:' address in the Apex outbound email message
3. Used the ID of the aforementioned Contact as the targetObjectId thereby enabling the whatId to be my object with the error details. The contact's Id was managed in a custom setting.