Both those ideas make sense, but the paperwork that came with the card from China says it is a power cord and for "optimum" performance it should be plugged in. Does not say what will happen if not plugged in, nor what the benefits are. And I cannot find an appropriate socket in the MB.
I did not build this PC but bought it from the store. It is a Dell I think. (my daughter is sleeping in the same room as the PC and if I wake her my life is forfeit.)
But as you say it works as it is, so I will probably just leave it.
I hate computers.