Unable to delete phone/email/etc using profiles

Analysis: Profiles can create and update entities, but currently cannot delete them. Since phone, email, etc are entities in the db and not really contact "fields" (although they are presented that way on the profile) clearing those fields on the profile will produce the unexpected result of doing nothing.
This issue is closely related to CRM-10128 because not every user should be allowed to delete data using profiles. So that issue must be addressed first.