The other main point is "has authority over".
If you take the "assist" line, you also would have to say that ATC can only "assist" you in avoiding collision in the ATZ. Therefore, if you are on the ILS and have been handed over to the Tower, although you are still in the clag, the Tower guy can only "assist" you in avoiding a collision with that 172 doing some scud-running just in front of you...