There is a custom brought by the Rema to leave an additional candle lit during the lighting of the Menorah at Hanukkah. Does this candle need to be lit for the sake of the mitzvah, or can one use a candle that is already lit (assuming all other necessary requirements are being fulfilled (e.g distinction, etc.))?
The point of the shamash is to have a light that is not lit for the sake of the mitzvah so that one is not "using" the chanukah lights. Thus it cannot be for the sake of the mitzvah.
Note that the Shamash is not lit for a mitzvah at all. That is, since it is lit only for the sake of not "using" the chanukah lights, it does not have a "mitzvah" and thus, any light left on in the room can be used. It is just easier and has become the custom to have a shamash attached to the menorah to ensure that this is done.
For example, if the electric light is left on in the room, that would be sufficient but it is possible that someone could forget and turn it off. Thus, the shamash attached to the menorah shows that you mean to have the "extra" light on.